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While studying the plume rise Gaussian Model , I came across following Improper integral which I was unable to solve :

$$\int_{0}^\infty \int_{-\infty}^{\infty} \mathrm{exp}[-((y^2+z^2)/2)]\,dy\,dz$$

The source I am referring says to apply the transformation

$y=r \cos θ$

$z=r \sin θ$

I am unable to do solve this integral. Please help me out.

Bill Cook
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Shashank Dwivedi
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  • You may find this interesting: https://math.stackexchange.com/questions/9286/proving-int-0-infty-mathrme-x2-dx-dfrac-sqrt-pi2 and this http://www.math.uconn.edu/~kconrad/blurbs/analysis/gaussianintegral.pdf – Olivier Oloa Apr 14 '17 at 14:03

1 Answers1

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You are transforming the integral to polar coordinates.

If $y=r\cos\theta$ and $z=r\sin\theta$, then $y^2+z^2=r^2$. Also, your region $z \geq 0$ (and $y$ anything) is the upper-half of the $yz$-plane. This is $r \geq 0$ and $0 \leq \theta \leq \pi$ in polar.

Don't forget the Jacobian ($J=r$). Therefore, your integral transforms to $$\int_0^\pi \int_0^\infty e^{-r^2/2} \,r\,dr\,d\theta = -\pi e^{-r^2/2}\Bigg|_0^\infty = \pi$$

Technically, there are issues with transforming your iterated integral to a double integral and then back to an iterated integral (in the new coordinate system). But everything works out since all integrals converge...and converge absolutely since you have a non-negative function.

Bill Cook
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