integration--Finding Principal Value

[clickbb08]

I'm supposed to show that the P.V. of the integral from (-infinity to infinity) of Cosx/(x^2+9)dx is (pi/3e^2). I don't understand how to go about these kinds of problems. I know that I will have an ISP at -3i and 3i.

 

[dwsmith]

I'm supposed to show that the P.V. of the integral from (-infinity to infinity) of Cosx/(x^2+9)dx is (pi/3e^2). I don't understand how to go about these kinds of problems. I know that I will have an ISP at -3i and 3i.

First it should be
$$
\int_{-\infty}^{\infty}\frac{\cos x}{x^2 + 9}dx = \frac{\pi}{3e^3}
$$

Let $f(z) = \dfrac{e^{iz}}{(z-3i)(z+3i)}$

Then
$$
\int_{-\infty}^{\infty}\frac{\cos x}{x^2 + 9}dx = 2\pi i\sum_{\text{UHP}}\text{Res}_{z=z_j}f(z) + \pi i\sum_{\text{real axis}}\text{Res}_{z=z_j}f(z)
$$

UHP = upper half plane