Contour Integral: confusion about cosine/sine

[outhsakotad]

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?

 

[outhsakotad]

Nevermind. I see. The associated term with sine is not odd in this case, so we cannot just use e^(2iz).