Basically I have this function for which I need to calculate the inverse Fourier transform $$ \hat{u}(\xi,z) = (-i)^n\sqrt{\frac{2\pi}{2a}}H_n\left(\frac{\xi}{\sqrt{2a}}\right)e^{-\xi^2/4a}e^{-i\xi^2z} $$ I need to calculate the value of $u(x,z)$ by IFT of the function above.
Now I have this identity basically, $$ \mathscr{F}(H_n(\sqrt{2a}x)e^{-ax^2}) =(-i)^n\sqrt{\frac{2\pi}{2a}}H_n\left(\frac{\xi}{\sqrt{2a}}\right)e^{-\xi^2/4a} $$ Thus I used the convolution theorem to break the function in two, mainly $$ \mathscr{F}^{-1}\left(\hat{u}(\xi,z)\right) = \mathscr{F}^{-1}\left((-i)^n\sqrt{\frac{2\pi}{2a}}H_n\left(\frac{\xi}{\sqrt{2a}}\right)e^{-\xi^2/4a}\right)\ast\mathscr{F}^{-1}\left(e^{-i\xi^2z}\right) $$ Thus I get $$ u(x,z) = \left(H_n\left(\sqrt{2a}x\right)e^{-ax^2}\right)\ast\left(\frac{e^{ix^2/4z}}{\sqrt{2iz}}\right) $$ I end up with the integral $$ u(x,z) = \int_{-\infty}^{\infty} d\tau \frac{1}{\sqrt{2iz}}H_n\left(\sqrt{2a}\tau\right)e^{-a\tau^2}e^{i(x-\tau)^2/4z} $$ I am not able to proceed further from here. I have two questions right now -
- Is this approach to solving this problem correct, or is there another way I should do this?
- If this is correct, how to I go about solving this integral? I have been stuck here for some time now.
Any help is appreciated. Original Problem
