# How to evaluate this integral?

1. Feb 9, 2012

### omoplata

1. The problem statement, all variables and given/known data

$$\int_{-\infty}^{\infty} \sin {( 2 \pi y )} \cdot e^{- \frac{(x-y)^2}{4 \nu t}} dy$$

x is between 0 and 1.
$\nu$ is a positive constant.
t is positive.

2. Relevant equations

3. The attempt at a solution

I tried substituting one variable for x-y, and integration by parts. So far, I've gotten nowhere. Can someone tell me which direction to go?

Thanks.

Edit: Also tried considering the Gaussian integral. But when I change the integration variable to get $e^{u^2}$, there's also a Sin term present. I tried to use integration by parts to get rid of that, but couldn't.

Last edited: Feb 9, 2012
2. Feb 9, 2012

### vela

Staff Emeritus
I haven't worked this out, but what I'd try is starting with u=x-y and expand the sin using a trig identity. Argue the sin u term will integrate to 0 and use
$$\cos \theta = \frac{e^{i\theta}+e^{-i\theta}}{2}$$ on the other term.