Difficult integral involving exponential

[zandria]

Homework Statement

I'm trying to verify the Fourier transform but am getting stuck on the integration. Here is the pair:
$$f(x) = e^{-ax^2}$$
$$\hat{f}(k) = \frac{1}{\sqrt{2a}}e^{-k^2/4a}$$
$$a>0$$

Homework Equations

I know that
$$\hat{f}(k)=\int_{-\infty}^{\infty}f(x)e^{ikx}dx$$

The Attempt at a Solution

So I have
$$\hat{f}(k)=\int_{-\infty}^{\infty}e^{-ax^2}e^{ikx}dx$$
$$\hat{f}(k)=\int_{-\infty}^{\infty}e^{-ax^2+ikx}dx$$

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.

 

[Cyosis]

You need to complete the square, which means:

$$-(ax^2+ikx)=-\left[(\alpha x+\beta)^2+\gamma \right]$$.

Find $\alpha,\beta$ and $\gamma$.

Edit1: it seems you have either listed $\hat{f}(k)$ wrong or the book where you got $\hat{f}(k)$ from is wrong, because the answer should be:

$$\hat{f}(k) = \sqrt{\frac{\pi}{a}}e^{-k^2/4a}$$

edit2: While making no difference to the final answer in this case, shouldn't it be $$\hat{f}(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}dx$$, note the minus sign.