calculate $\int_{-\infty}^{+\infty} \cos(at) e^{-bt^2} dt$

Question

Could someone please help me to calculate the integral of:

$$\int_{-\infty}^{+\infty} \cos (at) e^{-bt^2} dt.$$

a and b both real, b>0.

I have tried integration by parts, but I can't seem to simplify it to anything useful. Essentially, I would like to arrive at something that looks like: 7.4.6 here: textbook result

Answer 1

Do you have restrictions on 'a' and 'b'?

For example, they are real and > 0.

Otherwise, things are messy!

See here for details.

Answer 2

Hint:

Use the fact that $$\int_{-\infty}^\infty e^{iat- bt^2}\,dt = \sqrt{\frac{\pi}{b}} e^{-a^2/4b} $$ which is valid for $b>0$.

To derive this formula, complete the square in the exponent and then shift the integration contour a bit.