Square of the Error Function

Question

The author defines the probability integral as follows $$\Phi(z)=\frac{2}{\sqrt{\pi}}\int_0^z e^{-t^2}\mathrm{d}t$$. (S)he instructs the reader to derive the following integral representation of the square of the probability integral by transforming it to polar coordinates $$\Phi(z)^2=1-\frac{4}{\pi}\int_0^1\frac{\exp(-z^2(1+t^2))}{1+t^2}\mathrm{d}t$$. I am uncertain how transforming the integral from cartesian coordinates to polar coordinates would yield the integral representation. Can someone explain this to me? $$I^2=\frac{4}{\pi}\int_{0}^{z}\int_{0}^{z}e^{-(x^2+y^2)}dydx=-\frac{2}{\pi}\int_{0}^\frac{\pi}{2}\int_{0}^{z}-2re^{-r^2}drd\theta$$?

Answer 1

$\operatorname{erf}(z)^2$ can be expressed as an integral over the region $[0,z]\times[0,z]$. By symmetry, you can calculate half of its value by looking at the lower half triangle (i.e. with the additional bound $y\leq x$). I believe the correct coordinate transformation is $R=x^2+y^2, t=\frac yx$ (you could think of the former as $r^2$ and the latter as $\tan\theta$ in polar coordinates). Express the region in the new coordinates, calculate the Jacobian, and the result should follow.

Full solution: $\def\pdv#1#2{\frac{\partial #1}{\partial #2}}$

First, $erf^2(z)=\frac4\pi\int_{[0,z]^2}e^{-(x^2+y^2)}dxdy=\frac8\pi\int_{D}e^{-(x^2+y^2)}dxdy$, where $D=\{(x,y)\in[0,z]^2 \mid y\leq x\}$. Write $I=\int_{D}e^{-(x^2+y^2)}dxdy$.

The inverse of the Jacobian (since this is easier to calculate) is $\begin{pmatrix}\pdv Rx&\pdv Ry\\ \pdv tx&\pdv ty\end{pmatrix}$ with determinant $2(1+t^2)$, so its inverse has determinant $\frac1{2(1+t^2)}$. The triangular region $D$ becomes the region $\{(R,t)|0\leq t\leq1, 0\leq R\leq z^2(t^2+1)\}$. (You need to notice $\frac1{\cos^2\theta}=\tan^2\theta+1$.)

At last the calculation. $$I=\int_0^1\int_0^{z^2(t^2+1)}e^{-R}\cdot\frac1{2(1+t^2)}dRdt=\frac12\int_0^1(1-e^{-z^2(1+t^2)})\frac1{1+t^2}dt=\frac12(\frac\pi4-\int_0^1\frac{e^{-z^2(1+t^2)}}{1+t^2}dt),$$ so $$erf^2(z)=\frac8\pi I=1-\frac4\pi\int_0^1\frac{e^{-z^2(1+t^2)}}{1+t^2}dt$$ as desired.