The Shortest Period Of The Sum Of $2$ Functions With Coprime Commensurate Periods.

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Introduction

In the realm of periodic functions, understanding the properties of their sums is crucial for various applications in mathematics and physics. When dealing with two functions, $f$ and $g$, each with their own shortest period, $t_1$ and $t_2$, respectively, it is natural to wonder about the shortest period of their sum, $h = f + g$. Specifically, if $t_1 = 2$ and $t_2 = 3$, we may ask whether the shortest period of $h$ is indeed $6$. In this article, we will delve into the properties of periodic functions, explore the concept of coprime commensurate periods, and examine the relationship between the periods of $f$, $g$, and $h$.

Periodic Functions and Their Properties

A function $f(x)$ is said to be periodic with period $T$ if there exists a positive real number $T$ such that $f(x + T) = f(x)$ for all $x$ in the domain of $f$. The shortest period of a function is the smallest positive real number $T$ for which this property holds. Periodic functions are ubiquitous in mathematics and physics, appearing in the study of waves, vibrations, and other oscillatory phenomena.

Coprime Commensurate Periods

Two positive integers $m$ and $n$ are said to be coprime if their greatest common divisor (GCD) is $1$. In the context of periodic functions, two periods $t_1$ and $t_2$ are coprime if their GCD is $1$. When $t_1$ and $t_2$ are coprime, it means that the functions $f$ and $g$ have no common period, and their periods are relatively prime.

The Sum of Two Periodic Functions

Given two periodic functions $f$ and $g$ with periods $t_1$ and $t_2$, respectively, we can define their sum as $h = f + g$. The question arises: what is the shortest period of $h$? In general, the shortest period of $h$ is not simply the sum of the shortest periods of $f$ and $g$. To understand this, let's consider an example.

Example: Coprime Periods

Suppose we have two functions $f(x) = \sin(\frac{\pi x}{2})$ and $g(x) = \sin(\frac{\pi x}{3})$. The shortest periods of $f$ and $g$ are $t_1 = 2$ and $t_2 = 3$, respectively. Their sum is $h(x) = \sin(\frac{\pi x}{2}) + \sin(\frac{\pi x}{3})$. To find the shortest period of $h$, we need to find the smallest positive real number $T$ such that $h(x + T) = h(x)$ for all $x$.

Analyzing the Period of the Sum

To analyze the period of the sum $h$, we can use the following approach. Let $T$ be the shortest period of $h$. Then, for any $x$, we have:

$$h(x + T) = h(x)$$

Using the definition of $h$, we can rewrite this equation as:

$$\sin(\frac{\pi (x + T)}{2}) + \sin(\frac{\pi (x + T)}{3}) = \sin(\frac{\pi x}{2}) + \sin(\frac{\pi x}{3})$$

Expanding the sines, we get:

$$\sin(\frac{\pi x}{2} + \frac{\pi T}{2}) + \sin(\frac{\pi x}{3} + \frac{\pi T}{3}) = \sin(\frac{\pi x}{2}) + \sin(\frac{\pi x}{3})$$

Using the angle addition formula for sine, we can rewrite this equation as:

$$\sin(\frac{\pi x}{2})\cos(\frac{\pi T}{2}) + \cos(\frac{\pi x}{2})\sin(\frac{\pi T}{2}) + \sin(\frac{\pi x}{3})\cos(\frac{\pi T}{3}) + \cos(\frac{\pi x}{3})\sin(\frac{\pi T}{3}) = \sin(\frac{\pi x}{2}) + \sin(\frac{\pi x}{3})$$

Simplifying this equation, we get:

$$\sin(\frac{\pi x}{2})\left(\cos(\frac{\pi T}{2}) - 1\right) + \sin(\frac{\pi x}{3})\left(\cos(\frac{\pi T}{3}) - 1\right) = 0$$

This equation must hold for all $x$. In particular, if we choose $x = 0$, we get:

$$\left(\cos(\frac{\pi T}{2}) - 1\right) + \left(\cos(\frac{\pi T}{3}) - 1\right) = 0$$

Simplifying this equation, we get:

$$\cos(\frac{\pi T}{2}) + \cos(\frac{\pi T}{3}) = 2$$

Using the sum-to-product formula for cosine, we can rewrite this equation as:

$$2\cos(\frac{\pi T}{6})\cos(\frac{\pi T}{6}) = 2$$

Simplifying this equation, we get:

$$\cos(\frac{\pi T}{6}) = 1$$

This implies that:

$$\frac{\pi T}{6} = 2\pi k$$

for some integer $k$. Therefore, we have:

$$T = \frac{12\pi k}{\pi} = 12k$$

This means that the shortest period of $h$ is a multiple of $12$.

Conclusion

In conclusion, we have shown that the shortest period of the sum of two periodic functions with coprime commensurate periods is not necessarily the sum of their shortest periods. In fact, the shortest period of the sum is a multiple of the least common multiple of the shortest periods of the two functions. This result has important implications for the study of periodic functions and their properties.

References

• [1] A. Erdélyi, "Tables of Integral Transforms", McGraw-Hill, 1954.
• [2] E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis", Cambridge University Press, 1927.
• [3] H. M. Edwards, "Riemann's Zeta Function", Academic Press, 1974.

Additional Notes

• The result presented in this article is a special case of a more general result that can be obtained using the theory of Fourier series.
• The concept of coprime commensurate periods is a fundamental idea in the study of periodic functions and their properties.
• The shortest period of the sum of two periodic functions with coprime commensurate periods is a multiple of the least common multiple of their shortest periods.
  Q&A: The Shortest Period of the Sum of $2$ Functions with Coprime Commensurate Periods =====================================================================================

Q: What is the shortest period of the sum of two periodic functions with coprime commensurate periods?

A: The shortest period of the sum of two periodic functions with coprime commensurate periods is a multiple of the least common multiple of their shortest periods.

Q: What are coprime commensurate periods?

A: Two positive integers $m$ and $n$ are said to be coprime if their greatest common divisor (GCD) is $1$. In the context of periodic functions, two periods $t_1$ and $t_2$ are coprime if their GCD is $1$. When $t_1$ and $t_2$ are coprime, it means that the functions $f$ and $g$ have no common period, and their periods are relatively prime.

Q: How do you find the shortest period of the sum of two periodic functions?

A: To find the shortest period of the sum of two periodic functions, you can use the following approach:

1. Find the shortest periods of the two functions, $t_1$ and $t_2$.
2. Find the least common multiple (LCM) of $t_1$ and $t_2$.
3. The shortest period of the sum of the two functions is a multiple of the LCM.

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest positive integer that is a multiple of both numbers.

Q: How do you find the LCM of two numbers?

A: To find the LCM of two numbers, you can use the following approach:

1. List the multiples of each number.
2. Find the smallest multiple that is common to both lists.
3. The LCM is the smallest multiple that is common to both lists.

Q: Can you give an example of finding the shortest period of the sum of two periodic functions?

A: Suppose we have two functions $f(x) = \sin(\frac{\pi x}{2})$ and $g(x) = \sin(\frac{\pi x}{3})$. The shortest periods of $f$ and $g$ are $t_1 = 2$ and $t_2 = 3$, respectively. To find the shortest period of the sum of $f$ and $g$, we can use the following approach:

1. Find the LCM of $t_1$ and $t_2$, which is $6$.
2. The shortest period of the sum of $f$ and $g$ is a multiple of $6$.

Q: What are some applications of the shortest period of the sum of two periodic functions?

A: The shortest period of the sum of two periodic functions has applications in various fields, including:

• Signal processing: The shortest period of the sum of two periodic functions is used to analyze and process signals in various applications, such as image and audio processing.
• Control systems: The shortest period of the sum of two periodic functions is used to design and analyze control systems, such as feedback control systems.
• Electrical engineering: The shortest period of the sum of two periodic functions is used to analyze and design electrical circuits, such as filters and amplifiers.

Q: What are some common mistakes to avoid when finding the shortest period of the sum of two periodic functions?

A: Some common mistakes to avoid when finding the shortest period of the sum of two periodic functions include:

• Not finding the LCM of the shortest periods of the two functions.
• Not using the correct formula for the shortest period of the sum of two periodic functions.
• Not checking for coprime commensurate periods.

Q: How can I learn more about the shortest period of the sum of two periodic functions?

A: You can learn more about the shortest period of the sum of two periodic functions by:

• Reading books and articles on the subject.
• Taking online courses or attending workshops on signal processing and control systems.
• Practicing problems and exercises on the subject.
• Joining online communities and forums related to signal processing and control systems.