# Need help in complex integration

Can somebody help me with this.

## Homework Statement

$$\int_{-\infty}^{\infty} \, \frac{\sin{x}}{x} \, dx$$

## The Attempt at a Solution

1. Use complex numbers as there is a pole of order=0 at x=0
$$\int_{-\infty}^{\infty} \! f(x) \, dx = 2\pi\, i \sum_{res\, upper\, hp} {f(x)} \, + \pi\, i \sum_{res\, real\, axis} {f(x)}$$

which give 0 as the answer

2. Expand by sin(x) by Taylor series around 0 and multiply by x this gives a divergent series

Couldn't figure out which is correct?

Thanks

## The Attempt at a Solution

have you drawn the contour for this problem? the fact that there is a simple pole at x=0 (i.e. on the real axis) means you'll need a small semi-circular indent to take care of this.

also you can use Cauchy's residue theorem to get the integral round the whole contour then your probably going to need some sort of analysis with Jordan's lemma to eliminate the contributions from the parts of the contour not on the real line...

oh it will be easier to consider $\int_{-\infty}^{\infty} \frac{e^iz}{z} dz$ and then take the imaginary part of this at the end.

Thanks.
I tried the contour as in image
http://en.wikipedia.org/wiki/File:Contour_of_KKR.svg" [Broken]

taking the pole to z=0.

I used the following theorem

$$\textbf{Principal Value}\left\{\int_{-\infty}^{\infty} \! f(x) \, dx \right\} = 2\pi\, i \sum_{res\, upper\, hp} {f(x)} \, + \pi\, i \sum_{res\, real\, axis} {f(x)}$$

But there are no poles in upper half plane so the first sum is zero.

There's a pole at x=0 on real axis. Residue at x=0 is zero.

I'd really like to how u came to the below conclusion
$\int_{-\infty}^{\infty} \frac{e^iz}{z} dz$

is it by experience or is there any standard procedure u followed?

Last edited by a moderator:
Dick