Help with integral $\int_{-\infty}^\infty e^{iαx-α^2 t} dα$

Question

Solve by integration

$$\int_{-\infty}^\infty e^{-iαx-α^2 t} dα$$

solve by integration , this integral is - infinity to infinity and exponent value

Answer 1

This question has a short answer and a long one.

completing the square you have: $$ e^{−iαx−α^2t}=e^{-t(\alpha+\frac{ix}{2t})} \,e^{\frac{-x^2}{4t}} \\ \Rightarrow \int_{-\infty}^{\infty} e^{−iαx−α^2t}\, d \alpha=e^{\frac{-x^2}{4t}} \, \int_{-\infty}^{\infty}e^{-t(\alpha+\frac{ix}{2t})^2} \,d\alpha \\=e^{\frac{-x^2}{4t}} \, \int_{-\infty}^{\infty}e^{-t\alpha^2} \,d\alpha \\=e^{\frac{-x^2}{4t}} \sqrt{\frac{\pi}{t}} $$

But in the complete answer, we have to prove that

$$ \int_{-\infty}^{\infty}e^{-t(\alpha+b)^2} \,d\alpha=\int_{-\infty}^{\infty}e^{-t\alpha^2} \,d\alpha $$

For that we can just use a change of variable, but $b$ is imaginary and that means that the limits of integration would change to a line in the complex plane.

So to prove that we need another approach and what is commonly done is to use the fact $e^{z^2}$ is an entire function, so any closed line integral in the complex plane will be $0$. So you choose a rectangle whose segments parallel to the real line go from $-\infty$ to $\infty$ , and the other segments go from $0$ to some $ic$. Then you solve the 4 integrals for each segment and using that the sum of the 4 of them is $0$, you should get the desired result, that is, it is fine to complete the squares even if you have to deal with an imaginary term.

Answer 2

Although this isn't quite the Fourier transform of a Gaussian, this link helps illustrate @Mhenni's idea:

http://www4.ncsu.edu/~franzen/public_html/CH795Z/math/ft/gaussian.html

Answer 3

Hint: Complete the square in the argument of the exponential in terms of $\alpha$ and then use Gaussian integrals.

Added: Note that, you need to make a change of variables after completing the square. See here.