Exponential integral

Question

I have to evaluate $$\int_{-\infty}^\infty dx \int_{-\infty}^\infty dy\quad e^{-k(x^2+y^2+xy/4)}$$. I tried to transform it using $x=(\alpha+\beta)/\sqrt{2}$ and $y=(\alpha-\beta)/\sqrt{2}$. The integrand became $\dfrac{9\alpha^2+7\beta^2}{8}$ but what can be done about $dx\;dy$?

Answer 1

Why not to first complete the square $$x^2+y^2+\frac 14 xy=\left(y+\frac{x}{8}\right)^2+\frac{63 }{64}x^2$$ Let $z=y+\frac{x}{8}$ to face $$\int e^{-k \left(x^2+y^2+\frac{x y}{4}\right)}\,dy=e^{-\frac{63 k }{64}x^2} \int e^{-k z^2}\,dz$$ and you will have successively two gaussian integrals.