Let for a constant .

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Multiple integrals and change of variables

Problem

Let G(a)=∫−∞∞e−ax2 dxG(a) = \int_{-\infty}^{\infty} e^{-ax^2}\,\mathrm dx for a constant a>0a > 0. ((a)) Derive G(a)=π/aG(a)=\sqrt{\pi/a} without quoting the result. ((b)) Evaluate ∬e−(x2+xy+y2) dx dy\displaystyle\iint e^{-(x^2+xy+y^2)}\,\mathrm{d}x\,\mathrm{d}y over the whole xyxy plane.

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