Integral on complex plane of a gaussian times power

Question

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{x^2 R^2}{1+R^2} - \frac{y^2 }{1+R^2} - \frac{2 i x y R^2}{1+R^2}} d x d y $$ where $0<R<1$ and $k\geq 0$ is integer. Mathematica can solve the integral if I put an actual value for $k$, and evaluating it for $k=0,1,2,\dots$ it appears a recursive relation that suggests $$ I = \frac{(2k-1)!! \pi}{2^k} \left(\frac{\sqrt{1+R^2}}{R}\right)^{2k+1} $$ However, I am not able to prove it (I tried 2D-Fourier transform, Gaussian moments,...). Have you seen this formula around? Can you give me any hints o references?

Answer 1

Hint: Let $~\alpha=\dfrac{R^2}{1+R^2},~$ and $~J=\displaystyle\int_{\mathbb R^2}\exp\bigg[-\alpha~(x+iy)^2-y^2\bigg]~dx~dy.~$ Now, evaluate this

simpler expression, and see what happens as you differentiate repeatedly with regard to $\alpha$, a total

number of k times. :-)