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I want to find Fourier Transform of this function $f(t)=e^{-4t^2-4t-1}$, and hence Fourier Transform would take form:

$$(Ff)(x)=\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-4t^2-4t-1}e^{-ixt}dt$$

How can I conduct integration by parts for this integral?

Melina
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    If you know the Fourier transform of the gaussian, you just need to complete the square and do a shift in the variable. – mickep Dec 02 '15 at 19:24

1 Answers1

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Just write

$$e^{-4t^2 - 4t - 1} = e^{-(2t + 1)^2}$$

and try to evaluate this integral. It's easy because it's an Gaussian integral.

Enrico M.
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