Indented Contours Around Poles Of Order >1?
Introduction
Complex analysis is a branch of mathematics that deals with the study of complex numbers and their properties. One of the fundamental concepts in complex analysis is contour integration, which involves integrating a function over a closed curve in the complex plane. When computing a contour integral, it is possible to account for the contribution of a singularity on the contour in the sense of a Cauchy principal value. However, when dealing with poles of order greater than 1, the situation becomes more complex, and the use of indented contours becomes necessary.
What are Indented Contours?
Indented contours are a technique used in complex analysis to deal with poles of order greater than 1. A pole of order greater than 1 is a point on the contour where the function has a singularity, and the function can be written as a power series in a neighborhood of that point. The power series has a finite number of terms, and the function can be written as a sum of a finite number of terms, each of which is a power of the variable.
When dealing with a pole of order greater than 1, the contour integral can be written as a sum of two integrals, one of which is a contour integral around the pole, and the other of which is a contour integral around the pole with an indentation. The indentation is a small circle around the pole, and the contour integral around the pole with an indentation is called the indented contour integral.
Why are Indented Contours Necessary?
Indented contours are necessary when dealing with poles of order greater than 1 because the contour integral around the pole can be infinite. This is because the function has a singularity at the pole, and the contour integral around the pole can be written as an integral of the function over a small circle around the pole. The integral of the function over a small circle around the pole can be infinite if the function has a singularity at the pole.
The use of indented contours allows us to deal with this situation by introducing a small circle around the pole, and the contour integral around the pole with an indentation is called the indented contour integral. The indented contour integral is a contour integral around the pole with an indentation, and it can be written as a sum of two integrals, one of which is a contour integral around the pole, and the other of which is a contour integral around the pole with an indentation.
How to Compute Indented Contours?
To compute an indented contour, we need to follow these steps:
- Identify the pole: The first step is to identify the pole of order greater than 1. This can be done by analyzing the function and finding the point where the function has a singularity.
- Draw the contour: The next step is to draw the contour around the pole. The contour should be a closed curve that encloses the pole.
- Indent the contour: The contour should be indented by a small circle around the pole. The indentation should be small enough that it does not affect the contour integral around the pole.
- Compute the contour integral: The final step is to compute the contour integral around the pole with an indentation. This can be done by writing the function as a power series in a neighborhood of the pole and integrating the power series over the contour.
Example of Indented Contours
Let's consider an example of an indented contour. Suppose we have a function f(z) = 1/(z^2 - 1) and we want to compute the contour integral around the pole at z = 1.
The function f(z) = 1/(z^2 - 1) has a pole of order 2 at z = 1. To compute the contour integral around the pole, we need to indent the contour by a small circle around the pole.
The contour integral around the pole with an indentation can be written as:
∮ f(z) dz = ∮ f(z) dz + ∮ f(z) dz
where the first integral is the contour integral around the pole, and the second integral is the contour integral around the pole with an indentation.
The contour integral around the pole can be written as:
∮ f(z) dz = ∮ 1/(z^2 - 1) dz
The contour integral around the pole with an indentation can be written as:
∮ f(z) dz = ∮ 1/(z^2 - 1) dz
The contour integral around the pole with an indentation can be computed by writing the function as a power series in a neighborhood of the pole and integrating the power series over the contour.
Conclusion
Indented contours are a technique used in complex analysis to deal with poles of order greater than 1. The use of indented contours allows us to deal with the situation where the contour integral around the pole can be infinite. The contour integral around the pole with an indentation can be written as a sum of two integrals, one of which is a contour integral around the pole, and the other of which is a contour integral around the pole with an indentation.
The use of indented contours is necessary when dealing with poles of order greater than 1, and it allows us to compute the contour integral around the pole with an indentation. The contour integral around the pole with an indentation can be computed by writing the function as a power series in a neighborhood of the pole and integrating the power series over the contour.
References
- [1] Ahlfors, L. V. (1979). Complex Analysis. McGraw-Hill.
- [2] Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill.
- [3] Conway, J. B. (1995). Functions of One Complex Variable. Springer-Verlag.
Further Reading
- [1] Complex Analysis by Lars V. Ahlfors
- [2] Real and Complex Analysis by Walter Rudin
- [3] Functions of One Complex Variable by John B. Conway
Related Topics
- [1] Contour Integration
- [2] Singularity
- [3] Cauchy Principal Value
Indented Contours Around Poles of Order >1? =====================================================
Q&A
Q: What is the purpose of indented contours in complex analysis?
A: The purpose of indented contours is to deal with poles of order greater than 1 in complex analysis. When dealing with poles of order greater than 1, the contour integral can be infinite, and the use of indented contours allows us to compute the contour integral around the pole with an indentation.
Q: How do I identify the pole of order greater than 1?
A: To identify the pole of order greater than 1, you need to analyze the function and find the point where the function has a singularity. This can be done by writing the function as a power series in a neighborhood of the pole and identifying the point where the power series has a finite number of terms.
Q: What is the difference between a contour integral and an indented contour integral?
A: A contour integral is an integral of a function over a closed curve in the complex plane, while an indented contour integral is an integral of a function over a closed curve with an indentation around the pole.
Q: How do I compute the contour integral around the pole with an indentation?
A: To compute the contour integral around the pole with an indentation, you need to write the function as a power series in a neighborhood of the pole and integrate the power series over the contour.
Q: What is the significance of the indentation in the contour integral?
A: The indentation in the contour integral is necessary to deal with the situation where the contour integral around the pole can be infinite. The indentation allows us to compute the contour integral around the pole with an indentation.
Q: Can I use indented contours for any type of pole?
A: No, indented contours are only used for poles of order greater than 1. For poles of order 1, the contour integral can be computed using the Cauchy principal value.
Q: How do I know if I need to use an indented contour?
A: You need to use an indented contour if the function has a pole of order greater than 1 and the contour integral around the pole can be infinite.
Q: Can I use indented contours for functions with multiple poles?
A: Yes, you can use indented contours for functions with multiple poles. However, you need to indent the contour around each pole separately and compute the contour integral around each pole with an indentation.
Q: What are some common applications of indented contours?
A: Indented contours are commonly used in complex analysis to deal with poles of order greater than 1 in functions such as 1/(z^2 - 1) and 1/(z^3 - 1).
Q: How do I choose the size of the indentation?
A: The size of the indentation should be small enough that it does not affect the contour integral around the pole. A good rule of thumb is to choose the size of the indentation to be smaller than the distance between the pole and the nearest singularity.
Q: Can I use indented contours for functions with branch points?
A: No, indented contours are not used for functions with branch points. Branch points are points where the function has a branch cut, and the contour integral around the branch point can be infinite.
Conclusion
Indented contours are a technique used in complex analysis to deal with poles of order greater than 1. The use of indented contours allows us to compute the contour integral around the pole with an indentation, which is necessary when the contour integral around the pole can be infinite. The contour integral around the pole with an indentation can be computed by writing the function as a power series in a neighborhood of the pole and integrating the power series over the contour.
References
- [1] Ahlfors, L. V. (1979). Complex Analysis. McGraw-Hill.
- [2] Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill.
- [3] Conway, J. B. (1995). Functions of One Complex Variable. Springer-Verlag.
Further Reading
- [1] Complex Analysis by Lars V. Ahlfors
- [2] Real and Complex Analysis by Walter Rudin
- [3] Functions of One Complex Variable by John B. Conway
Related Topics
- [1] Contour Integration
- [2] Singularity
- [3] Cauchy Principal Value