# Need method to solve contour integral

• freechus9
In summary, Jason is trying to solve a nasty integral and is not sure if the method he is using is correct. He is trying to solve the equation \int^{\infty}_{-\infty}\frac{e^{ikx}dk}{\sqrt{k^2 + a} - (b\pm i\lambda)} but has trouble justifying the multiplication of the integrand by "1."
freechus9
Hello, I am working on a problem and I have to solve a nasty integral. The problem is that I am not sure if the method I am using is correct.

The integral I need to solve is:

$\int$$^{\infty}_{-\infty}$$\frac{e^{ikx}dk}{\sqrt{k^2 + a} - (b\pm i\lambda)}$

At this point, I have tried multiplying the top and bottom by $\sqrt{k^2 + a} + (b\pm i\lambda)$ and continuing forth by splitting into partial fractions and using the Cauchy residue theorem, but I am not sure if this is a valid way to solve this.

If anyone has any insight I would be more than indebted! Thank you so much!

It can work, as long as you are taking into account the fact that your integrand is multiple valued. You have branch points at $\pm i a$. The typical approach is to define a single valued "branch" of the integrand with a corresponding branch cut, then do the normal contour integral thing. But in this case, the path you use to close your contour has to contend with the branch cut - you will most likely end up with integrals along the two sides of the branch cut in addition to your residue. Now these new integrals may be easier than your original integral, or recognizable as some special function or something, or they may be beasts unto themselves.

Most complex analysis books should show examples of this sort of thing.

hope that helps ...

jason

Actually, I am already familiar with those methods for contour integration. The main problem I am having is whether or not it is legal to multiply the integrand by a specific value of "1," namely,

$\frac{\sqrt{k^2 + a} + (b\pm i\lambda)}{ \sqrt{k^2 + a} + (b\pm i\lambda) }$.

This would allow me to rewrite the integrand as a sum over two first order polynomials in the denominator, which is then solvable by the Cauchy residue method. However, it is the multiplication by "1" which I am having trouble justifying. Hope that is clearer.

Thanks!

## 1. What is a contour integral?

A contour integral is a type of line integral that is used to calculate the total change of a function over a closed curve or contour. It is often used in complex analysis to evaluate the integral of a complex-valued function.

## 2. Why is a contour integral useful?

A contour integral is useful because it allows us to calculate the total change of a function over a closed curve, which can be difficult to do using traditional methods. It is also a powerful tool in complex analysis and has many applications in physics, engineering, and mathematics.

## 3. What is the need for a method to solve contour integrals?

Contour integrals can be challenging to solve using traditional integration techniques, especially when dealing with complex functions. Therefore, a specialized method is needed to accurately and efficiently solve contour integrals.

## 4. What are some common methods used to solve contour integrals?

There are several methods that can be used to solve contour integrals, including the Cauchy integral theorem, Cauchy's integral formula, and the residue theorem. These methods involve using complex analysis techniques to evaluate the integral over a contour.

## 5. What are some tips for solving contour integrals?

Some tips for solving contour integrals include choosing an appropriate contour, understanding the properties of the function being integrated, and using symmetry to simplify the integral. It is also essential to have a strong understanding of complex analysis and integration techniques.

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