Contour Integral: confusion about cosine/sine

In summary, the speaker was trying to integrate cos(2x)/(x-i*pi) from -inf to inf. They converted cos(2x) to e^(2iz) and used a large semicircle in the UHP as their contour. They encountered a simple pole at z=i*pi and determined the residue to be e^(-2pi). However, their final answer was half of what it should be. They eventually realized that the term associated with sine was not odd in this case, so they could not just use e^(2iz).
  • #1
outhsakotad
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Homework Statement


I am to integrate cos(2x)/(x-i*pi) from -inf to inf


Homework Equations





The Attempt at a Solution


The problem I'm having is this:

I write cos(2x) in exponential form, e^(2iz), so f(z) = e^(2iz)/(z-i*pi). I choose a large semicircle in the UHP as my contour. At large R, the integral over the arc goes to 0.

There is a simple pole at z=i*pi. The residue at the simple pole is then lim z-->i*pi [e^(2iz)] = e^(-2pi), so I get the integral to be 2*pi*i*e^(-2pi). But the answer is pi*i*e^(-2pi), half of what I get.

This answer can be gotten if one writes cos(2z) in the UHP as 1/2(e^(2iz)), but other problems with sine or cosine I have successfully solved writing sine or cosine without using this factor of 1/2. What am I missing here?
 
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  • #2
Nevermind. I see. The associated term with sine is not odd in this case, so we cannot just use e^(2iz).
 

1. What is a contour integral?

A contour integral is a type of complex integral that involves integrating along a path in the complex plane. It is used to calculate the area under a curve in the complex plane, similar to how a regular integral is used to calculate the area under a curve in the real plane.

2. How is cosine/sine used in a contour integral?

Both cosine and sine are used as part of the parametrization of the contour in a contour integral. They help define the path along which the integral is taken. The specific choice of cosine or sine depends on the direction of the contour, either clockwise or counterclockwise, respectively.

3. Why is there confusion about using cosine/sine in contour integrals?

The confusion arises because in regular integrals, cosine and sine are used as part of the trigonometric substitution method. However, in contour integrals, they are used as part of the parametrization of the contour and have a different purpose.

4. Can I use other functions besides cosine/sine in a contour integral?

Yes, depending on the specific contour and function being integrated, other functions such as exponential, logarithmic, or polynomial functions may be used in the parametrization. The choice of function is based on making the integral easier to evaluate.

5. How do I know which contour to choose for a contour integral?

The choice of contour depends on the specific problem and the function being integrated. Generally, a contour is chosen to avoid singularities of the function and to make the integral easier to evaluate. It is also important to ensure that the contour is closed and does not intersect itself.

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