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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

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    Sorry. done now2012-09-04

2 Answers 2

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

For example, they are real and > 0.

Otherwise, things are messy!

See here for details.

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    a and b are both real; and b>0. Hmm.. I should have written this question up a bit better. I apologise.2012-09-04
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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.

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    Actually, you overlooked that minus sign.2013-03-14