How to calculate $\iint_{\mathbb{R}^2} \exp\left(-x^2-y^2+ixy-ix-iy\right)\,dx\,dy$?

Question

How do you calculate the following? $$\iint_{\mathbb{R}^2} \exp\left(-x^2-y^2+ixy-ix-iy\right)\,dx\,dy$$ It might be solved by representing $x$ and $y$ with a pair of other variables but I have no idea.

Answer 1

This is a Gaussian double-integral $$ I ~:=~\iint_{\mathbb{R}^2} \! dx~dy~ e^{-S(x,y)} ,\tag{1} $$ where $$ S(x,y)~:=~x^2+y^2-ixy+ix+iy. \tag{2}$$ One method is to change coordinates $x^{\pm}=\frac{x\pm y}{\sqrt{2}}$ and complete the square. However computationally it is easier to just use the method of steepest descent which yields exact results for Gaussian integrals. The stationary point is $$(x_0,y_0)~=~ \left(\frac{1}{1+2i},\frac{1}{1+2i}\right). \tag{3}$$ The integral therefore becomes $$ I~=~\frac{2\pi}{\sqrt{\det(S^{\prime\prime}(x_0,y_0))}} e^{-S(x_0,y_0)} ~=~\frac{2\pi}{\sqrt{5}} \exp\left(\frac{1}{i-2}\right).\tag{4}$$