Contour integral with absolute value

In summary, the conversation discusses the computation of the integral \int_{-\infty}^{\infty}\frac{\textrm{sech}\hspace{0.1cm} x}{x^{2}-2|x|+1}dx via contour integration. One method suggested is to split the domain into two parts, \int_{-\infty}^{0}\frac{\textrm{sech}\hspace{0.1cm} x}{x^{2}+2x+1}dx and \int_{0}^{\infty}\frac{\textrm{sech}\hspace{0.1cm} x}{x^{2}-2x+1}dx
  • #1
hunt_mat
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Suppose I want to compute tthe integral:
[tex]
\int_{-\infty}^{\infty}\frac{\textrm{sech}\hspace{0.1cm} x}{x^{2}-2|x|+1}dx
[/tex]
Can I compute this integral via contour integration? The only way that I have thought of is to split up the domain:
[tex]
\int_{-\infty}^{\infty}\frac{\textrm{sech}\hspace{0.1cm} x}{x^{2}-2|x|+1}dx=\int_{-\infty}^{0}\frac{\textrm{sech}\hspace{0.1cm} x}{x^{2}+2x+1}dx+\int_{0}^{\infty}\frac{\textrm{sech}\hspace{0.1cm} x}{x^{2}-2x+1}dx
[/tex]
Is this the best way I can go about things for is there a better way?
 
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  • #2
So far OK, but you have singularities at |x|=1.
 
  • #3
I am aware of the singularities, this was just an example. The other integral doesn't have singularities, I just wanted to get my ideas across.
 

FAQ: Contour integral with absolute value

1. What is a contour integral with absolute value?

A contour integral with absolute value is an integral that involves the absolute value of a function along a given contour. It is used in complex analysis to evaluate the behavior of complex functions along a particular path.

2. How is a contour integral with absolute value different from a regular contour integral?

A regular contour integral involves integrating a complex function along a given path, while a contour integral with absolute value involves taking the absolute value of the function before integrating. This can change the behavior and value of the integral.

3. What is the significance of the absolute value in a contour integral?

The absolute value in a contour integral is used to capture the overall behavior of a complex function along a given path. It helps to take into account both the magnitude and direction of the function, rather than just the direction in a regular contour integral.

4. How is a contour integral with absolute value used in real-world applications?

A contour integral with absolute value is commonly used in physics and engineering to calculate the work done by a vector field along a specific path. It is also used in the study of fluid dynamics and electromagnetic fields.

5. What are some techniques for evaluating a contour integral with absolute value?

There are several techniques for evaluating a contour integral with absolute value, including using the Cauchy integral formula, the residue theorem, and the method of successive approximations. These techniques involve breaking down the integral into smaller, more manageable parts and using properties of complex functions to evaluate them.

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