- #1

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## Homework Statement

I'm trying to verify the Fourier transform but am getting stuck on the integration. Here is the pair:

[tex]f(x) = e^{-ax^2}[/tex]

[tex]\hat{f}(k) = \frac{1}{\sqrt{2a}}e^{-k^2/4a}[/tex]

[tex]a>0[/tex]

## Homework Equations

I know that

[tex]\hat{f}(k)=\int_{-\infty}^{\infty}f(x)e^{ikx}dx[/tex]

## The Attempt at a Solution

So I have

[tex]\hat{f}(k)=\int_{-\infty}^{\infty}e^{-ax^2}e^{ikx}dx[/tex]

[tex]\hat{f}(k)=\int_{-\infty}^{\infty}e^{-ax^2+ikx}dx[/tex]

I tried using integration by parts and I'm not sure that's the right way to go. If it is I'm not sure how to go about it without getting a more complicated integral.