Integrate Faddeeva function

Question

I came across this integration in my studies.

$\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$

It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I am unsure how to integrate this equation since it contains this function. I searched online for quite a while but could not find any information pertaining to my problem. Can someone point me in the right direction to solve this integral?

Answer 1

For the Faddeeva function

$$F(z)=e^{-z^2}\, \text{erfc}(-i z)\tag{1}$$

the integral

$$I=\int\limits_{-\infty}^{\infty} |F(z)|^2 \, dz\tag{2}$$

can be evaluated using the Plancherel theorem

$$\int\limits_{-\infty}^{\infty} |F(z)|^2 \, dz=\int\limits_{-\infty}^{\infty} |\hat{F}(\omega)|^2 \, d\omega\tag{3}$$

where

$$\hat{F}(\omega)=\mathcal{F}_z[F(z)](\omega)=\int\limits_{-\infty}^{\infty} F(z)\, e^{-i 2 \pi \omega z} \, dz=\sqrt{\pi}\, e^{-\pi^2 \omega^2}\, (\text{sgn}(\omega)+1)\tag{4}$$

is the Fourier transform of $F(z)$ leading to the result

$$I=\int\limits_{-\infty}^{\infty} \left|\sqrt{\pi}\, e^{-\pi^2 \omega^2}\, (\text{sgn}(\omega)+1)\right| ^2 \, d\omega=\sqrt{2 \pi}\tag{5}.$$