Calculate $\iint\limits_{\mathbb{R}^2} (x^2+y^2)^p e^{-x^2-y^2} d x d y$.

Question

I want to calculate an integral $$ \iint\limits_{\mathbb{R}^2} (x^2+y^2)^p e^{-x^2-y^2} d x d y, $$ using gamma-functions.

Answer 1

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Use first polar coordinates , so your integral is

$$\int_0^{2\pi}\int_0^\infty r^{2p}e^{-r^2}rdr\,d\theta=2\pi\int_0^\infty r^{2p+1}e^{-r^2}dr\;\;(**)$$

and now a new substitution: $\;t:=r^2\implies dt=2r\,dr\implies dr=\frac{dt}{2\sqrt t}\;$:

$$(**)=2\pi\int_0^\infty t^{p+1/2}e^{-t}\frac{dt}{2\sqrt t}=\pi\,\Gamma(p+1)$$

...as long as $\;\Gamma(p+1)\;$ is defined , of course.