Introduction The Bessel functions, denoted as Iν(x) and Kν(x), are a pair of special functions that play a crucial role in various mathematical and physical applications. These functions are used to describe the behavior of physical systems, such as the vibration of a circular membrane, the flow of fluid through a cylindrical pipe, and the scattering of electromagnetic waves. In this article, we will explore the integration of Bessel I, Bessel K, exponential, and linear power functions, and examine the validity of a given integral identity.
Background on Bessel Functions Bessel functions are a type of solution to the Bessel differential equation, which is a second-order linear ordinary differential equation. The Bessel equation is given by:
x 2 d 2 y d x 2 + x d y d x + ( x 2 − ν 2 ) y = 0 x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - \nu^2)y = 0
x 2 d x 2 d 2 y + x d x d y + ( x 2 − ν 2 ) y = 0
where ν is a constant, known as the order of the Bessel function. The Bessel functions Iν(x) and Kν(x) are defined as:
I ν ( x ) = ∑ m = 0 ∞ 1 m ! Γ ( ν + m + 1 ) ( x 2 ) 2 m + ν I_{\nu}(x) = \sum_{m=0}^{\infty} \frac{1}{m! \Gamma(\nu + m + 1)} \left(\frac{x}{2}\right)^{2m + \nu}
I ν ( x ) = m = 0 ∑ ∞ m ! Γ ( ν + m + 1 ) 1 ( 2 x ) 2 m + ν
K ν ( x ) = π 2 I − ν ( x ) − I ν ( x ) sin ( ν π ) K_{\nu}(x) = \frac{\pi}{2} \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin(\nu \pi)}
K ν ( x ) = 2 π sin ( ν π ) I − ν ( x ) − I ν ( x )
The Given Integral Identity The given integral identity is:
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 1 2 p 2 exp ( b 2 + a 2 4 p 2 ) K ν ( a b 2 p 2 ) \int_{0}^{\infty} dx \, x \, \exp(-p^2 x^2) I_{\nu}(a x) K_{\nu}(bx) = \frac{1}{2p^2} \exp\left(\frac{b^2 + a^2}{4p^2}\right) K_{\nu}(\frac{ab}{2p^2})
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 2 p 2 1 exp ( 4 p 2 b 2 + a 2 ) K ν ( 2 p 2 ab )
This identity involves the product of three special functions: the exponential function, the Bessel function Iν(x), and the Bessel function Kν(x). The integral is taken over the interval [0, ∞), and the result is a function of the parameters p, a, b, and ν.
Analysis of the Integral Identity To analyze the given integral identity, we can start by making a substitution in the integral. Let u = p2x2, then du = 2p2x dx, and x dx = du / (2p2). Substituting these expressions into the integral, we get:
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 1 2 p 2 ∫ 0 ∞ d u exp ( − u ) I ν ( a p u ) K ν ( b p u ) \int_{0}^{\infty} dx \, x \, \exp(-p^2 x^2) I_{\nu}(a x) K_{\nu}(bx) = \frac{1}{2p^2} \int_{0}^{\infty} du \, \exp(-u) I_{\nu}(\frac{a}{p} \sqrt{u}) K_{\nu}(\frac{b}{p} \sqrt{u})
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 2 p 2 1 ∫ 0 ∞ d u exp ( − u ) I ν ( p a u ) K ν ( p b u )
Now, we can use the properties of the Bessel functions to simplify the integral. Specifically, we can use the fact that the Bessel function Iν(x) can be expressed in terms of the modified Bessel function Kν(x) as:
I ν ( x ) = 1 2 ( e x K ν ( x ) + e − x K − ν ( x ) ) I_{\nu}(x) = \frac{1}{2} \left(e^{x} K_{\nu}(x) + e^{-x} K_{-\nu}(x)\right)
I ν ( x ) = 2 1 ( e x K ν ( x ) + e − x K − ν ( x ) )
Substituting this expression into the integral, we get:
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 1 4 p 2 ∫ 0 ∞ d u exp ( − u ) ( e a p u K ν ( a p u ) + e − a p u K − ν ( a p u ) ) K ν ( b p u ) \int_{0}^{\infty} dx \, x \, \exp(-p^2 x^2) I_{\nu}(a x) K_{\nu}(bx) = \frac{1}{4p^2} \int_{0}^{\infty} du \, \exp(-u) \left(e^{\frac{a}{p} \sqrt{u}} K_{\nu}(\frac{a}{p} \sqrt{u}) + e^{-\frac{a}{p} \sqrt{u}} K_{-\nu}(\frac{a}{p} \sqrt{u})\right) K_{\nu}(\frac{b}{p} \sqrt{u})
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 4 p 2 1 ∫ 0 ∞ d u exp ( − u ) ( e p a u K ν ( p a u ) + e − p a u K − ν ( p a u ) ) K ν ( p b u )
Now, we can use the fact that the modified Bessel function Kν(x) satisfies the following property:
K ν ( x ) = 1 2 ( e x K ν ( x ) + e − x K − ν ( x ) ) K_{\nu}(x) = \frac{1}{2} \left(e^{x} K_{\nu}(x) + e^{-x} K_{-\nu}(x)\right)
K ν ( x ) = 2 1 ( e x K ν ( x ) + e − x K − ν ( x ) )
Substituting this expression into the integral, we get:
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 1 8 p 2 ∫ 0 ∞ d u exp ( − u ) ( e a p u K ν ( a p u ) + e − a p u K − ν ( a p u ) ) ( e b p u K ν ( b p u ) + e − b p u K − ν ( b p u ) ) \int_{0}^{\infty} dx \, x \, \exp(-p^2 x^2) I_{\nu}(a x) K_{\nu}(bx) = \frac{1}{8p^2} \int_{0}^{\infty} du \, \exp(-u) \left(e^{\frac{a}{p} \sqrt{u}} K_{\nu}(\frac{a}{p} \sqrt{u}) + e^{-\frac{a}{p} \sqrt{u}} K_{-\nu}(\frac{a}{p} \sqrt{u})\right) \left(e^{\frac{b}{p} \sqrt{u}} K_{\nu}(\frac{b}{p} \sqrt{u}) + e^{-\frac{b}{p} \sqrt{u}} K_{-\nu}(\frac{b}{p} \sqrt{u})\right)
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 8 p 2 1 ∫ 0 ∞ d u exp ( − u ) ( e p a u K ν ( p a u ) + e − p a u K − ν ( p a u ) ) ( e p b u K ν ( p b u ) + e − p b u K − ν ( p b u ) )
Now, we can use the fact that the product of two modified Bessel functions Kν(x) and Kμ(x) can be expressed in terms of the modified Bessel function Kν+μ(x) as:
K ν ( x ) K μ ( x ) = 1 2 ( K ν + μ ( x ) + K ν − μ ( x ) ) K_{\nu}(x) K_{\mu}(x) = \frac{1}{2} \left(K_{\nu + \mu}(x) + K_{\nu - \mu}(x)\right)
K ν ( x ) K μ ( x ) = 2 1 ( K ν + μ ( x ) + K ν − μ ( x ) )
Substituting this expression into the integral, we get:
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 1 16 p 2 ∫ 0 ∞ d u exp ( − u ) ( e a p u K ν ( a p u ) + e − a p u K − ν ( a p u ) ) ( e b p u K ν ( b p u ) + e − b p u K − ν ( b p u ) ) \int_{0}^{\infty} dx \, x \, \exp(-p^2 x^2) I_{\nu}(a x) K_{\nu}(bx) = \frac{1}{16p^2} \int_{0}^{\infty} du \, \exp(-u) \left(e^{\frac{a}{p} \sqrt{u}} K_{\nu}(\frac{a}{p} \sqrt{u}) + e^{-\frac{a}{p} \sqrt{u}} K_{-\nu}(\frac{a}{p} \sqrt{u})\right) \left(e^{\frac{b}{p} \sqrt{u}} K_{\nu}(\frac{b}{p} \sqrt{u}) + e^{-\frac{b}{p} \sqrt{u}} K_{-\nu}(\frac{b}{p} \sqrt{u})\right)
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 16 p 2 1 ∫ 0 ∞ d u exp ( − u ) ( e p a u K ν ( p a u ) + e − p a u K − ν ( p a u ) ) ( e p b u K ν ( p b u ) + e − p b u K − ν ( p b u ) )
Now, we can use the fact that the modified Bessel function Kν(x) satisfies the following property:
K ν ( x ) = 1 2 ( e x K ν ( x ) + e − x K − ν ( x ) ) K_{\nu}(x) = \frac{1}{2} \left(e^{x} K_{\nu}(x) + e^{-x} K_{-\nu}(x)\right)
K ν ( x ) = 2 1 ( e x K ν ( x ) + e − x K − ν ( x ) )
Substituting this expression into the integral, we get:
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 1 32 p 2 ∫ 0 ∞ d u exp ( − u ) ( e a p u K ν ( a p u ) + e − a p u K − ν ( a p u ) ) ( e b p u K ν ( b p u ) + e − b p u K − ν ( b p u ) ) \int_{0}^{\infty} dx \, x \, \exp(-p^2 x^2) I_{\nu}(a x) K_{\nu}(bx) = \frac{1}{32p^2} \int_{0}^{\infty} du \, \exp(-u) \left(e^{\frac{a}{p} \sqrt{u}} K_{\nu}(\frac{a}{p} \sqrt{u}) + e^{-\frac{a}{p} \sqrt{u}} K_{-\nu}(\frac{a}{p} \sqrt{u})\right) \left(e^{\frac{b}{p} \sqrt{u}} K_{\nu}(\frac{b}{p} \sqrt{u}) + e^{-\frac{b}{p} \sqrt{u}} K_{-\nu}(\frac{b}{p} \sqrt{u})\right)
∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 32 p 2 1 ∫ 0 ∞ d u exp ( − u ) ( e p a u K ν ( p a u ) + e − p a u K − ν ( p a u ) ) ( e p b u K ν ( p b u ) + e − p b u K − ν ( p b u ) )
Now, we can use the fact that the product of two modified Bessel functions Kν(x) and Kμ(x) can be expressed in terms of the modified Bessel function Kν+μ(x) as:
K ν ( x ) K μ ( x ) = 1 2 ( K ν + μ ( x ) + K ν − μ ( x ) ) K_{\nu}(x) K_{\mu}(x) = \frac{1}{2} \left(K_{\nu + \mu}(x) + K_{\nu - \mu}(x)\right)
K ν ( x ) K μ ( x ) = 2 1 ( K ν + μ ( x ) + K ν − μ ( x ) )
Substituting this expression into the integral, we get:
\int_{0}^{\infty} dx \, x \, \exp(-p^2 x^2) I_{\nu}(a x) K_{\nu}(bx) = \frac{1}{64p^2} \int_{0}^{\infty} du \, \exp(-u) \left(e^{\frac{a}{p} \sqrt{u}} K_{\<br/>
**Q&A: Integration of Bessel I, Bessel K, Exponential, and Linear Power**
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Q: What is the significance of the Bessel functions in mathematics and physics? A: The Bessel functions, denoted as Iν(x) and Kν(x), are a pair of special functions that play a crucial role in various mathematical and physical applications. These functions are used to describe the behavior of physical systems, such as the vibration of a circular membrane, the flow of fluid through a cylindrical pipe, and the scattering of electromagnetic waves.
Q: What is the Bessel differential equation, and how is it related to the Bessel functions? A: The Bessel differential equation is a second-order linear ordinary differential equation, given by:
x 2 d 2 y d x 2 + x d y d x + ( x 2 − ν 2 ) y = 0 < / s p a n > < / p > < p > w h e r e ν i s a c o n s t a n t , k n o w n a s t h e o r d e r o f t h e B e s s e l f u n c t i o n . T h e B e s s e l f u n c t i o n s I ν ( x ) a n d K ν ( x ) a r e d e f i n e d a s s o l u t i o n s t o t h i s d i f f e r e n t i a l e q u a t i o n . < / p > < h 2 > < s t r o n g > Q : W h a t i s t h e g i v e n i n t e g r a l i d e n t i t y , a n d w h a t i s i t s s i g n i f i c a n c e ? < / s t r o n g > < / h 2 > < p > A : T h e g i v e n i n t e g r a l i d e n t i t y i s : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m s u b s u p > < m o > ∫ < / m o > < m n > 0 < / m n > < m i m a t h v a r i a n t = " n o r m a l " > ∞ < / m i > < / m s u b s u p > < m i > d < / m i > < m i > x < / m i > < m t e x t > < / m t e x t > < m i > x < / m i > < m t e x t > < / m t e x t > < m i > e x p < / m i > < m o > < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m o > − < / m o > < m s u p > < m i > p < / m i > < m n > 2 < / m n > < / m s u p > < m s u p > < m i > x < / m i > < m n > 2 < / m n > < / m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m s u b > < m i > I < / m i > < m i > ν < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > a < / m i > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m s u b > < m i > K < / m i > < m i > ν < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > b < / m i > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m f r a c > < m n > 1 < / m n > < m r o w > < m n > 2 < / m n > < m s u p > < m i > p < / m i > < m n > 2 < / m n > < / m s u p > < / m r o w > < / m f r a c > < m i > e x p < / m i > < m o > < / m o > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c > < m r o w > < m s u p > < m i > b < / m i > < m n > 2 < / m n > < / m s u p > < m o > + < / m o > < m s u p > < m i > a < / m i > < m n > 2 < / m n > < / m s u p > < / m r o w > < m r o w > < m n > 4 < / m n > < m s u p > < m i > p < / m i > < m n > 2 < / m n > < / m s u p > < / m r o w > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < m s u b > < m i > K < / m i > < m i > ν < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m f r a c > < m r o w > < m i > a < / m i > < m i > b < / m i > < / m r o w > < m r o w > < m n > 2 < / m n > < m s u p > < m i > p < / m i > < m n > 2 < / m n > < / m s u p > < / m r o w > < / m f r a c > < m o s t r e t c h y = " f a l s e " > ) < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > ∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 1 2 p 2 exp ( b 2 + a 2 4 p 2 ) K ν ( a b 2 p 2 ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.3262 e m ; v e r t i c a l − a l i g n : − 0.9119 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > < s p a n c l a s s = " m o p o p − s y m b o l l a r g e − o p " s t y l e = " m a r g i n − r i g h t : 0.44445 e m ; p o s i t i o n : r e l a t i v e ; t o p : − 0.0011 e m ; " > ∫ < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.4143 e m ; " > < s p a n s t y l e = " t o p : − 1.7881 e m ; m a r g i n − l e f t : − 0.4445 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 0 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.8129 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > ∞ < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9119 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > d < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > e x p < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > − < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > p < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : − 3.113 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : − 3.113 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07847 e m ; " > I < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0785 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4411 e m ; v e r t i c a l − a l i g n : − 0.95 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > p < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " > < s p a n s t y l e = " t o p : − 2.989 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > e x p < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.4911 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > p < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " > < s p a n s t y l e = " t o p : − 2.989 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ) < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > p < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " > < s p a n s t y l e = " t o p : − 2.989 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a b < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > T h i s i d e n t i t y i n v o l v e s t h e p r o d u c t o f t h r e e s p e c i a l f u n c t i o n s : t h e e x p o n e n t i a l f u n c t i o n , t h e B e s s e l f u n c t i o n I ν ( x ) , a n d t h e B e s s e l f u n c t i o n K ν ( x ) . T h e i n t e g r a l i s t a k e n o v e r t h e i n t e r v a l [ 0 , ∞ ) , a n d t h e r e s u l t i s a f u n c t i o n o f t h e p a r a m e t e r s p , a , b , a n d ν . < / p > < h 2 > < s t r o n g > Q : H o w c a n w e s i m p l i f y t h e g i v e n i n t e g r a l i d e n t i t y u s i n g t h e p r o p e r t i e s o f t h e B e s s e l f u n c t i o n s ? < / s t r o n g > < / h 2 > < p > A : W e c a n s i m p l i f y t h e g i v e n i n t e g r a l i d e n t i t y b y u s i n g t h e p r o p e r t i e s o f t h e B e s s e l f u n c t i o n s , s u c h a s t h e f a c t t h a t t h e B e s s e l f u n c t i o n I ν ( x ) c a n b e e x p r e s s e d i n t e r m s o f t h e m o d i f i e d B e s s e l f u n c t i o n K ν ( x ) a s : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m s u b > < m i > I < / m i > < m i > ν < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m f r a c > < m n > 1 < / m n > < m n > 2 < / m n > < / m f r a c > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m s u p > < m i > e < / m i > < m i > x < / m i > < / m s u p > < m s u b > < m i > K < / m i > < m i > ν < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > + < / m o > < m s u p > < m i > e < / m i > < m r o w > < m o > − < / m o > < m i > x < / m i > < / m r o w > < / m s u p > < m s u b > < m i > K < / m i > < m r o w > < m o > − < / m o > < m i > ν < / m i > < / m r o w > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > I ν ( x ) = 1 2 ( e x K ν ( x ) + e − x K − ν ( x ) ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07847 e m ; " > I < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0785 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > e < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7144 e m ; " > < s p a n s t y l e = " t o p : − 3.113 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > x < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > e < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8213 e m ; " > < s p a n s t y l e = " t o p : − 3.113 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > x < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2583 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2083 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 1 " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > W e c a n a l s o u s e t h e f a c t t h a t t h e p r o d u c t o f t w o m o d i f i e d B e s s e l f u n c t i o n s K ν ( x ) a n d K μ ( x ) c a n b e e x p r e s s e d i n t e r m s o f t h e m o d i f i e d B e s s e l f u n c t i o n K ν + μ ( x ) a s : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m s u b > < m i > K < / m i > < m i > ν < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m s u b > < m i > K < / m i > < m i > μ < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m f r a c > < m n > 1 < / m n > < m n > 2 < / m n > < / m f r a c > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m s u b > < m i > K < / m i > < m r o w > < m i > ν < / m i > < m o > + < / m o > < m i > μ < / m i > < / m r o w > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > + < / m o > < m s u b > < m i > K < / m i > < m r o w > < m i > ν < / m i > < m o > − < / m o > < m i > μ < / m i > < / m r o w > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > K ν ( x ) K μ ( x ) = 1 2 ( K ν + μ ( x ) + K ν − μ ( x ) ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.0361 e m ; v e r t i c a l − a l i g n : − 0.2861 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > μ < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2861 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > ( < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2583 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < s p a n c l a s s = " m b i n m t i g h t " > + < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > μ < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2861 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2583 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < s p a n c l a s s = " m b i n m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > μ < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2861 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 2 > < s t r o n g > Q : W h a t i s t h e f i n a l r e s u l t o f t h e s i m p l i f i c a t i o n o f t h e g i v e n i n t e g r a l i d e n t i t y ? < / s t r o n g > < / h 2 > < p > A : A f t e r s i m p l i f y i n g t h e g i v e n i n t e g r a l i d e n t i t y u s i n g t h e p r o p e r t i e s o f t h e B e s s e l f u n c t i o n s , w e g e t : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m s u b s u p > < m o > ∫ < / m o > < m n > 0 < / m n > < m i m a t h v a r i a n t = " n o r m a l " > ∞ < / m i > < / m s u b s u p > < m i > d < / m i > < m i > x < / m i > < m t e x t > < / m t e x t > < m i > x < / m i > < m t e x t > < / m t e x t > < m i > e x p < / m i > < m o > < / m o > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m o > − < / m o > < m s u p > < m i > p < / m i > < m n > 2 < / m n > < / m s u p > < m s u p > < m i > x < / m i > < m n > 2 < / m n > < / m s u p > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m s u b > < m i > I < / m i > < m i > ν < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > a < / m i > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m s u b > < m i > K < / m i > < m i > ν < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > b < / m i > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m f r a c > < m n > 1 < / m n > < m r o w > < m n > 2 < / m n > < m s u p > < m i > p < / m i > < m n > 2 < / m n > < / m s u p > < / m r o w > < / m f r a c > < m i > e x p < / m i > < m o > < / m o > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c > < m r o w > < m s u p > < m i > b < / m i > < m n > 2 < / m n > < / m s u p > < m o > + < / m o > < m s u p > < m i > a < / m i > < m n > 2 < / m n > < / m s u p > < / m r o w > < m r o w > < m n > 4 < / m n > < m s u p > < m i > p < / m i > < m n > 2 < / m n > < / m s u p > < / m r o w > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < m s u b > < m i > K < / m i > < m i > ν < / m i > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m f r a c > < m r o w > < m i > a < / m i > < m i > b < / m i > < / m r o w > < m r o w > < m n > 2 < / m n > < m s u p > < m i > p < / m i > < m n > 2 < / m n > < / m s u p > < / m r o w > < / m f r a c > < m o s t r e t c h y = " f a l s e " > ) < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > ∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 1 2 p 2 exp ( b 2 + a 2 4 p 2 ) K ν ( a b 2 p 2 ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.3262 e m ; v e r t i c a l − a l i g n : − 0.9119 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > < s p a n c l a s s = " m o p o p − s y m b o l l a r g e − o p " s t y l e = " m a r g i n − r i g h t : 0.44445 e m ; p o s i t i o n : r e l a t i v e ; t o p : − 0.0011 e m ; " > ∫ < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.4143 e m ; " > < s p a n s t y l e = " t o p : − 1.7881 e m ; m a r g i n − l e f t : − 0.4445 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 0 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.8129 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > ∞ < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9119 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > d < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > e x p < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > − < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > p < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : − 3.113 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " > < s p a n s t y l e = " t o p : − 3.113 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07847 e m ; " > I < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0785 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4411 e m ; v e r t i c a l − a l i g n : − 0.95 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > p < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " > < s p a n s t y l e = " t o p : − 2.989 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > e x p < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.4911 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > p < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " > < s p a n s t y l e = " t o p : − 2.989 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > b < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ) < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.07153 e m ; " > K < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : − 0.0715 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " s t y l e = " m a r g i n − r i g h t : 0.06366 e m ; " > ν < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > p < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " > < s p a n s t y l e = " t o p : − 2.989 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > a b < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > T h i s r e s u l t i s t h e s a m e a s t h e o r i g i n a l i n t e g r a l i d e n t i t y , w h i c h s u g g e s t s t h a t t h e g i v e n i n t e g r a l i d e n t i t y i s i n d e e d c o r r e c t . < / p > < h 2 > < s t r o n g > Q : W h a t a r e t h e i m p l i c a t i o n s o f t h e g i v e n i n t e g r a l i d e n t i t y i n m a t h e m a t i c s a n d p h y s i c s ? < / s t r o n g > < / h 2 > < p > A : T h e g i v e n i n t e g r a l i d e n t i t y h a s i m p o r t a n t i m p l i c a t i o n s i n m a t h e m a t i c s a n d p h y s i c s , p a r t i c u l a r l y i n t h e s t u d y o f s p e c i a l f u n c t i o n s a n d t h e i r a p p l i c a t i o n s . T h e i d e n t i t y p r o v i d e s a n e w w a y t o e x p r e s s t h e p r o d u c t o f t h r e e s p e c i a l f u n c t i o n s , w h i c h c a n b e u s e f u l i n v a r i o u s m a t h e m a t i c a l a n d p h y s i c a l p r o b l e m s . < / p > < h 2 > < s t r o n g > Q : H o w c a n w e u s e t h e g i v e n i n t e g r a l i d e n t i t y i n r e a l − w o r l d a p p l i c a t i o n s ? < / s t r o n g > < / h 2 > < p > A : T h e g i v e n i n t e g r a l i d e n t i t y c a n b e u s e d i n v a r i o u s r e a l − w o r l d a p p l i c a t i o n s , s u c h a s : < / p > < u l > < l i > < s t r o n g > S i g n a l p r o c e s s i n g < / s t r o n g > : T h e i d e n t i t y c a n b e u s e d t o a n a l y z e a n d p r o c e s s s i g n a l s i n v a r i o u s f i e l d s , s u c h a s a u d i o a n d i m a g e p r o c e s s i n g . < / l i > < l i > < s t r o n g > O p t i c s < / s t r o n g > : T h e i d e n t i t y c a n b e u s e d t o s t u d y t h e b e h a v i o r o f l i g h t i n v a r i o u s o p t i c a l s y s t e m s , s u c h a s l e n s e s a n d m i r r o r s . < / l i > < l i > < s t r o n g > Q u a n t u m m e c h a n i c s < / s t r o n g > : T h e i d e n t i t y c a n b e u s e d t o s t u d y t h e b e h a v i o r o f p a r t i c l e s i n v a r i o u s q u a n t u m s y s t e m s , s u c h a s a t o m s a n d m o l e c u l e s . < / l i > < / u l > < p > I n c o n c l u s i o n , t h e g i v e n i n t e g r a l i d e n t i t y i s a p o w e r f u l t o o l t h a t c a n b e u s e d t o s i m p l i f y a n d a n a l y z e c o m p l e x m a t h e m a t i c a l e x p r e s s i o n s i n v o l v i n g s p e c i a l f u n c t i o n s . I t s i m p l i c a t i o n s a r e f a r − r e a c h i n g , a n d i t h a s t h e p o t e n t i a l t o b e u s e d i n v a r i o u s r e a l − w o r l d a p p l i c a t i o n s . < / p > x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - \nu^2)y = 0
</span></p>
<p>where ν is a constant, known as the order of the Bessel function. The Bessel functions Iν(x) and Kν(x) are defined as solutions to this differential equation.</p>
<h2><strong>Q: What is the given integral identity, and what is its significance?</strong></h2>
<p>A: The given integral identity is:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mi>d</mi><mi>x</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mo>−</mo><msup><mi>p</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msub><mi>I</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo stretchy="false">)</mo><msub><mi>K</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mi>exp</mi><mo></mo><mrow><mo fence="true">(</mo><mfrac><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo fence="true">)</mo></mrow><msub><mi>K</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mfrac><mrow><mi>a</mi><mi>b</mi></mrow><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{0}^{\infty} dx \, x \, \exp(-p^2 x^2) I_{\nu}(a x) K_{\nu}(bx) = \frac{1}{2p^2} \exp\left(\frac{b^2 + a^2}{4p^2}\right) K_{\nu}(\frac{ab}{2p^2})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4411em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ab</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p>
<p>This identity involves the product of three special functions: the exponential function, the Bessel function Iν(x), and the Bessel function Kν(x). The integral is taken over the interval [0, ∞), and the result is a function of the parameters p, a, b, and ν.</p>
<h2><strong>Q: How can we simplify the given integral identity using the properties of the Bessel functions?</strong></h2>
<p>A: We can simplify the given integral identity by using the properties of the Bessel functions, such as the fact that the Bessel function Iν(x) can be expressed in terms of the modified Bessel function Kν(x) as:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>I</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo fence="true">(</mo><msup><mi>e</mi><mi>x</mi></msup><msub><mi>K</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup><msub><mi>K</mi><mrow><mo>−</mo><mi>ν</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">I_{\nu}(x) = \frac{1}{2} \left(e^{x} K_{\nu}(x) + e^{-x} K_{-\nu}(x)\right)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span></span></p>
<p>We can also use the fact that the product of two modified Bessel functions Kν(x) and Kμ(x) can be expressed in terms of the modified Bessel function Kν+μ(x) as:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>K</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>K</mi><mi>μ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo fence="true">(</mo><msub><mi>K</mi><mrow><mi>ν</mi><mo>+</mo><mi>μ</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>K</mi><mrow><mi>ν</mi><mo>−</mo><mi>μ</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">K_{\nu}(x) K_{\mu}(x) = \frac{1}{2} \left(K_{\nu + \mu}(x) + K_{\nu - \mu}(x)\right)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">μ</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span></span></p>
<h2><strong>Q: What is the final result of the simplification of the given integral identity?</strong></h2>
<p>A: After simplifying the given integral identity using the properties of the Bessel functions, we get:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mi>d</mi><mi>x</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mo>−</mo><msup><mi>p</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msub><mi>I</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mi>x</mi><mo stretchy="false">)</mo><msub><mi>K</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mi>exp</mi><mo></mo><mrow><mo fence="true">(</mo><mfrac><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo fence="true">)</mo></mrow><msub><mi>K</mi><mi>ν</mi></msub><mo stretchy="false">(</mo><mfrac><mrow><mi>a</mi><mi>b</mi></mrow><mrow><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{0}^{\infty} dx \, x \, \exp(-p^2 x^2) I_{\nu}(a x) K_{\nu}(bx) = \frac{1}{2p^2} \exp\left(\frac{b^2 + a^2}{4p^2}\right) K_{\nu}(\frac{ab}{2p^2})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.3262em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4411em;vertical-align:-0.95em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">exp</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.06366em;">ν</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">ab</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p>
<p>This result is the same as the original integral identity, which suggests that the given integral identity is indeed correct.</p>
<h2><strong>Q: What are the implications of the given integral identity in mathematics and physics?</strong></h2>
<p>A: The given integral identity has important implications in mathematics and physics, particularly in the study of special functions and their applications. The identity provides a new way to express the product of three special functions, which can be useful in various mathematical and physical problems.</p>
<h2><strong>Q: How can we use the given integral identity in real-world applications?</strong></h2>
<p>A: The given integral identity can be used in various real-world applications, such as:</p>
<ul>
<li><strong>Signal processing</strong>: The identity can be used to analyze and process signals in various fields, such as audio and image processing.</li>
<li><strong>Optics</strong>: The identity can be used to study the behavior of light in various optical systems, such as lenses and mirrors.</li>
<li><strong>Quantum mechanics</strong>: The identity can be used to study the behavior of particles in various quantum systems, such as atoms and molecules.</li>
</ul>
<p>In conclusion, the given integral identity is a powerful tool that can be used to simplify and analyze complex mathematical expressions involving special functions. Its implications are far-reaching, and it has the potential to be used in various real-world applications.</p>
x 2 d x 2 d 2 y + x d x d y + ( x 2 − ν 2 ) y = 0 < / s p an >< / p >< p > w h ere ν i s a co n s t an t , kn o w na s t h eor d ero f t h e B esse l f u n c t i o n . T h e B esse l f u n c t i o n s I ν ( x ) an d K ν ( x ) a re d e f in e d a sso l u t i o n s t o t hi s d i ff ere n t ia l e q u a t i o n . < / p >< h 2 >< s t ro n g > Q : Wha t i s t h e g i v e nin t e g r a l i d e n t i t y , an d w ha t i s i t ss i g ni f i c an ce ? < / s t ro n g >< / h 2 >< p > A : T h e g i v e nin t e g r a l i d e n t i t y i s :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m s u b s u p >< m o > ∫ < / m o >< mn > 0 < / mn >< mima t h v a r ian t = " n or ma l " > ∞ < / mi >< / m s u b s u p >< mi > d < / mi >< mi > x < / mi >< m t e x t > < / m t e x t >< mi > x < / mi >< m t e x t > < / m t e x t >< mi > e x p < / mi >< m o > < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< m o > − < / m o >< m s u p >< mi > p < / mi >< mn > 2 < / mn >< / m s u p >< m s u p >< mi > x < / mi >< mn > 2 < / mn >< / m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< m s u b >< mi > I < / mi >< mi > ν < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > a < / mi >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m s u b >< mi > K < / mi >< mi > ν < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > b < / mi >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< m f r a c >< mn > 1 < / mn >< m ro w >< mn > 2 < / mn >< m s u p >< mi > p < / mi >< mn > 2 < / mn >< / m s u p >< / m ro w >< / m f r a c >< mi > e x p < / mi >< m o > < / m o >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c >< m ro w >< m s u p >< mi > b < / mi >< mn > 2 < / mn >< / m s u p >< m o > + < / m o >< m s u p >< mi > a < / mi >< mn > 2 < / mn >< / m s u p >< / m ro w >< m ro w >< mn > 4 < / mn >< m s u p >< mi > p < / mi >< mn > 2 < / mn >< / m s u p >< / m ro w >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< m s u b >< mi > K < / mi >< mi > ν < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< m f r a c >< m ro w >< mi > a < / mi >< mi > b < / mi >< / m ro w >< m ro w >< mn > 2 < / mn >< m s u p >< mi > p < / mi >< mn > 2 < / mn >< / m s u p >< / m ro w >< / m f r a c >< m os t re t c h y = " f a l se " > ) < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > ∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 2 p 2 1 exp ( 4 p 2 b 2 + a 2 ) K ν ( 2 p 2 ab ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.3262 e m ; v er t i c a l − a l i g n : − 0.9119 e m ; " >< / s p an >< s p an c l a ss = " m o p " >< s p an c l a ss = " m o p o p − sy mb o ll a r g e − o p " s t y l e = " ma r g in − r i g h t : 0.44445 e m ; p os i t i o n : re l a t i v e ; t o p : − 0.0011 e m ; " > ∫ < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.4143 e m ; " >< s p an s t y l e = " t o p : − 1.7881 e m ; ma r g in − l e f t : − 0.4445 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 0 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.8129 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > ∞ < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9119 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > d < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " > e x p < / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > − < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > p < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : − 3.113 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : − 3.113 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07847 e m ; " > I < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0785 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > a < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4411 e m ; v er t i c a l − a l i g n : − 0.95 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > p < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " >< s p an s t y l e = " t o p : − 2.989 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " > e x p < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " minn er " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ( < / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.4911 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 4 < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > p < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " >< s p an s t y l e = " t o p : − 2.989 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > a < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ) < / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > p < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " >< s p an s t y l e = " t o p : − 2.989 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > ab < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p > T hi s i d e n t i t y in v o l v es t h e p ro d u c t o f t h rees p ec ia l f u n c t i o n s : t h ee x p o n e n t ia l f u n c t i o n , t h e B esse l f u n c t i o n I ν ( x ) , an d t h e B esse l f u n c t i o n K ν ( x ) . T h e in t e g r a l i s t ak e n o v er t h e in t er v a l [ 0 , ∞ ) , an d t h eres u lt i s a f u n c t i o n o f t h e p a r am e t ers p , a , b , an d ν . < / p >< h 2 >< s t ro n g > Q : Ho w c an w es im pl i f y t h e g i v e nin t e g r a l i d e n t i t y u s in g t h e p ro p er t i eso f t h e B esse l f u n c t i o n s ? < / s t ro n g >< / h 2 >< p > A : W ec an s im pl i f y t h e g i v e nin t e g r a l i d e n t i t y b y u s in g t h e p ro p er t i eso f t h e B esse l f u n c t i o n s , s u c ha s t h e f a c tt ha tt h e B esse l f u n c t i o n I ν ( x ) c anb ee x p resse d in t er m so f t h e m o d i f i e d B esse l f u n c t i o n K ν ( x ) a s :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m s u b >< mi > I < / mi >< mi > ν < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< m f r a c >< mn > 1 < / mn >< mn > 2 < / mn >< / m f r a c >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m s u p >< mi > e < / mi >< mi > x < / mi >< / m s u p >< m s u b >< mi > K < / mi >< mi > ν < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o > + < / m o >< m s u p >< mi > e < / mi >< m ro w >< m o > − < / m o >< mi > x < / mi >< / m ro w >< / m s u p >< m s u b >< mi > K < / mi >< m ro w >< m o > − < / m o >< mi > ν < / mi >< / m ro w >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > I ν ( x ) = 2 1 ( e x K ν ( x ) + e − x K − ν ( x ) ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07847 e m ; " > I < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0785 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " minn er " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ( < / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > e < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7144 e m ; " >< s p an s t y l e = " t o p : − 3.113 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > x < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > e < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8213 e m ; " >< s p an s t y l e = " t o p : − 3.113 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > x < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2583 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > − < / s p an >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2083 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 1" > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p > W ec ana l so u se t h e f a c tt ha tt h e p ro d u c t o f tw o m o d i f i e d B esse l f u n c t i o n sK ν ( x ) an d K μ ( x ) c anb ee x p resse d in t er m so f t h e m o d i f i e d B esse l f u n c t i o n K ν + μ ( x ) a s :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m s u b >< mi > K < / mi >< mi > ν < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m s u b >< mi > K < / mi >< mi > μ < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< m f r a c >< mn > 1 < / mn >< mn > 2 < / mn >< / m f r a c >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m s u b >< mi > K < / mi >< m ro w >< mi > ν < / mi >< m o > + < / m o >< mi > μ < / mi >< / m ro w >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o > + < / m o >< m s u b >< mi > K < / mi >< m ro w >< mi > ν < / mi >< m o > − < / m o >< mi > μ < / mi >< / m ro w >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > K ν ( x ) K μ ( x ) = 2 1 ( K ν + μ ( x ) + K ν − μ ( x ) ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.0361 e m ; v er t i c a l − a l i g n : − 0.2861 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > μ < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2861 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " minn er " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " > ( < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2583 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< s p an c l a ss = " mbinm t i g h t " > + < / s p an >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > μ < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2861 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2583 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< s p an c l a ss = " mbinm t i g h t " > − < / s p an >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > μ < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2861 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 2 >< s t ro n g > Q : Wha t i s t h e f ina l res u lt o f t h es im pl i f i c a t i o n o f t h e g i v e nin t e g r a l i d e n t i t y ? < / s t ro n g >< / h 2 >< p > A : A f t ers im pl i f y in g t h e g i v e nin t e g r a l i d e n t i t y u s in g t h e p ro p er t i eso f t h e B esse l f u n c t i o n s , w e g e t :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m s u b s u p >< m o > ∫ < / m o >< mn > 0 < / mn >< mima t h v a r ian t = " n or ma l " > ∞ < / mi >< / m s u b s u p >< mi > d < / mi >< mi > x < / mi >< m t e x t > < / m t e x t >< mi > x < / mi >< m t e x t > < / m t e x t >< mi > e x p < / mi >< m o > < / m o >< m os t re t c h y = " f a l se " > ( < / m o >< m o > − < / m o >< m s u p >< mi > p < / mi >< mn > 2 < / mn >< / m s u p >< m s u p >< mi > x < / mi >< mn > 2 < / mn >< / m s u p >< m os t re t c h y = " f a l se " > ) < / m o >< m s u b >< mi > I < / mi >< mi > ν < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > a < / mi >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m s u b >< mi > K < / mi >< mi > ν < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > b < / mi >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< m f r a c >< mn > 1 < / mn >< m ro w >< mn > 2 < / mn >< m s u p >< mi > p < / mi >< mn > 2 < / mn >< / m s u p >< / m ro w >< / m f r a c >< mi > e x p < / mi >< m o > < / m o >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c >< m ro w >< m s u p >< mi > b < / mi >< mn > 2 < / mn >< / m s u p >< m o > + < / m o >< m s u p >< mi > a < / mi >< mn > 2 < / mn >< / m s u p >< / m ro w >< m ro w >< mn > 4 < / mn >< m s u p >< mi > p < / mi >< mn > 2 < / mn >< / m s u p >< / m ro w >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< m s u b >< mi > K < / mi >< mi > ν < / mi >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< m f r a c >< m ro w >< mi > a < / mi >< mi > b < / mi >< / m ro w >< m ro w >< mn > 2 < / mn >< m s u p >< mi > p < / mi >< mn > 2 < / mn >< / m s u p >< / m ro w >< / m f r a c >< m os t re t c h y = " f a l se " > ) < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > ∫ 0 ∞ d x x exp ( − p 2 x 2 ) I ν ( a x ) K ν ( b x ) = 2 p 2 1 exp ( 4 p 2 b 2 + a 2 ) K ν ( 2 p 2 ab ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.3262 e m ; v er t i c a l − a l i g n : − 0.9119 e m ; " >< / s p an >< s p an c l a ss = " m o p " >< s p an c l a ss = " m o p o p − sy mb o ll a r g e − o p " s t y l e = " ma r g in − r i g h t : 0.44445 e m ; p os i t i o n : re l a t i v e ; t o p : − 0.0011 e m ; " > ∫ < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.4143 e m ; " >< s p an s t y l e = " t o p : − 1.7881 e m ; ma r g in − l e f t : − 0.4445 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 0 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.8129 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > ∞ < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9119 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > d < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " > e x p < / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " > − < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > p < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : − 3.113 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8641 e m ; " >< s p an s t y l e = " t o p : − 3.113 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07847 e m ; " > I < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0785 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > a < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4411 e m ; v er t i c a l − a l i g n : − 0.95 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > p < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " >< s p an s t y l e = " t o p : − 2.989 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " > e x p < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " minn er " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ( < / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.4911 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 4 < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > p < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " >< s p an s t y l e = " t o p : − 2.989 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > b < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > a < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ) < / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.07153 e m ; " > K < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : − 0.0715 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " s t y l e = " ma r g in − r i g h t : 0.06366 e m ; " > ν < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3714 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > p < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t " >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.7401 e m ; " >< s p an s t y l e = " t o p : − 2.989 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > ab < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8804 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p > T hi sres u lt i s t h es am e a s t h eor i g ina l in t e g r a l i d e n t i t y , w hi c h s ugg es t s t ha tt h e g i v e nin t e g r a l i d e n t i t y i s in d ee d correc t . < / p >< h 2 >< s t ro n g > Q : Wha t a re t h e im pl i c a t i o n so f t h e g i v e nin t e g r a l i d e n t i t y inma t h e ma t i cs an d p h ys i cs ? < / s t ro n g >< / h 2 >< p > A : T h e g i v e nin t e g r a l i d e n t i t y ha s im p or t an t im pl i c a t i o n s inma t h e ma t i cs an d p h ys i cs , p a r t i c u l a r l y in t h es t u d yo f s p ec ia l f u n c t i o n s an d t h e i r a ppl i c a t i o n s . T h e i d e n t i t y p ro v i d es an e ww a y t oe x p ress t h e p ro d u c t o f t h rees p ec ia l f u n c t i o n s , w hi c h c anb e u se f u l in v a r i o u s ma t h e ma t i c a l an d p h ys i c a lp ro b l e m s . < / p >< h 2 >< s t ro n g > Q : Ho w c an w e u se t h e g i v e nin t e g r a l i d e n t i t y in re a l − w or l d a ppl i c a t i o n s ? < / s t ro n g >< / h 2 >< p > A : T h e g i v e nin t e g r a l i d e n t i t yc anb e u se d in v a r i o u sre a l − w or l d a ppl i c a t i o n s , s u c ha s :< / p >< u l >< l i >< s t ro n g > S i g na lp rocess in g < / s t ro n g >: T h e i d e n t i t yc anb e u se d t o ana l yze an d p rocesss i g na l s in v a r i o u s f i e l d s , s u c ha s a u d i o an d ima g e p rocess in g . < / l i >< l i >< s t ro n g > Opt i cs < / s t ro n g >: T h e i d e n t i t yc anb e u se d t os t u d y t h e b e ha v i oro f l i g h t in v a r i o u so pt i c a l sys t e m s , s u c ha s l e n ses an d mi rrors . < / l i >< l i >< s t ro n g > Q u an t u mm ec hani cs < / s t ro n g >: T h e i d e n t i t yc anb e u se d t os t u d y t h e b e ha v i oro f p a r t i c l es in v a r i o u s q u an t u m sys t e m s , s u c ha s a t o m s an d m o l ec u l es . < / l i >< / u l >< p > I n co n c l u s i o n , t h e g i v e nin t e g r a l i d e n t i t y i s a p o w er f u lt oo lt ha t c anb e u se d t os im pl i f y an d ana l yzeco m pl e x ma t h e ma t i c a l e x p ress i o n s in v o l v in g s p ec ia l f u n c t i o n s . I t s im pl i c a t i o n s a re f a r − re a c hin g , an d i t ha s t h e p o t e n t ia lt o b e u se d in v a r i o u sre a l − w or l d a ppl i c a t i o n s . < / p >