Double Gaussian Integral

Calculus Level 5

∫ − ∞ ∞ ∫ − ∞ ∞ e − ( x 2 + x y + y 2 ) d x d y = a π b \displaystyle \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} e^{-\left( x^2+xy+y^2 \right)} dxdy = \frac{a\pi}{\sqrt b}

Evaluate a + b a+b where b b is square free.


The answer is 5.

Aman Rajput by Aman Rajput , 25, India

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1 solution

Mark Hennings
May 11, 2018

Note the standard integral ∫ − ∞ ∞ e − u x 2 d x = π u \int_{-\infty}^\infty e^{-ux^2}\,dx \; = \; \sqrt{\tfrac{\pi}{u}} Then ∫ − ∞ ∞ e − x 2 − x y − y 2 d x = ∫ − ∞ ∞ e − ( x + 1 2 y ) 2 − 3 4 y 2 d x = π e − 3 4 y 2 \int_{-\infty}^\infty e^{-x^2-xy-y^2}\,dx \; = \; \int_{-\infty}^\infty e^{-(x+\frac12y)^2 - \frac34y^2}\,dx \; = \; \sqrt{\pi}e^{-\frac34y^2} so that the double integral is ∫ ∞ ∞ π e − 3 4 y 2 d y = 2 π 3 \int_{\infty}^\infty \sqrt{\pi} e^{-\frac34y^2}\,dy \; = \; \frac{2\pi}{\sqrt{3}} making the answer 2 + 3 = 5 2+3=\boxed{5} .

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