Inverse Fourier Transform of $F(\xi)=\frac{e^{-\frac{\xi^2}8}}{2^\frac{1}{4}}$

Question

I'm studying Fourier tranforms for the first time and I'm having trouble calculating this inverse. I think there might be some integration trick I don't know of to solve this.

$$F(\xi)=\frac{e^{-\frac{\xi^2}8}}{2^\frac{1}{4}}$$

I need to find the inverse transform of this.

My definition of inverse is

$$f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty F(\xi)e^{i\xi x}d\xi$$

Any hints would be appreciated.

Answer 1

HINT

Note that $\exp\left(-\frac{\xi^2}8\right)\exp(i\xi x)=\exp\left(-\frac{\xi^2}8+i\xi x\right)$ and complete the square. Then utilize the well-known Gaussian Intergral, i.e. $\int\limits_{-\infty}^\infty\exp(-x^2){\rm d}x=\sqrt{\pi}$, to get rid of the integral and to obtain the inverse function $f(x)$.