- Thread starter
- #1

#### quantaentangled

##### New member

- Apr 2, 2012

- 9

I am new to the site, so I thought this would be a good time to post an interesting integral I ran across that I am having a time with. It is a miscellaneous problem in Schaum's Outline of Complex Variables, #86 in ch. 7. I have been self-teaching a little CA when I get time and this one is a little more challenging than others I have encountered.

\(\displaystyle \int_{0}^{\infty}\frac{1}{(4x^{2}+\pi^{2})\cosh(x)}dx=\frac{\ln(2)}{2\pi}\)

Does anyone have a good idea of how to approach this one?.

At first, I thought perhaps a rectangular contour may do the trick, but now I am thinking perhaps a semicircular one may be in order.

It would appear there is a double pole at \(\displaystyle \frac{\pi i}{2}\).

Using the Laurent series, I think the residue is \(\displaystyle \frac{-i}{4\pi^{2}}\)

Which gives \(\displaystyle 2\pi i\left(\frac{-i}{4\pi^{2}}\right)=\frac{1}{2\pi}\)

But, how to get that ln(2)?. Maybe use \(\displaystyle \cos(ix)\) in some respect?.