The context of
$$\int_{-\frac{\pi}{2}}^ {\frac{\pi}{2}}\text{erf}\left(\frac{\sqrt 2 R\cos\theta}{\sigma}\right)\text d\theta$$
is it came up whilst integrating the Rayleigh distribution function over an off-centered circle of radius $R$:
$$\int_{-\frac{\pi}{2}}^ {\frac{\pi}{2}}\int_0^{2R\cos\theta}\frac{r}{\sigma^2}\exp\left(-\frac{r^2}{2\sigma^2}\right)r\text dr\text d\theta.$$
Since I’ve never worked with integrals of special functions such as the error function, I’d appreciate any help.
What’s $\int_{-\frac{\pi}{2}}^ {\frac{\pi}{2}}\text{erf}\left(\frac{\sqrt 2 R\cos\theta}{\sigma}\right)\text d\theta$
definite integralserror functionintegration
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Best Answer
By differentiation wrt R, integration wrt $\theta$ $$\frac{2}{\sigma }\ \sqrt{\frac{2}{\pi }}\quad \int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} \ \ e^{-\frac{2 R^2 \cos ^2(\theta )}{\sigma ^2}} \ \cos (\theta )\, d\theta $$
and integration wrt R Mathematica yields $$\int \frac{2}{R}\ e^{-\frac{2 R^2}{\sigma ^2}} \text{erfi}\left(\frac{\sqrt{2} R}{\sigma }\right) \, dR=\frac{4}{\sigma } \sqrt{\frac{2}{\pi }}\ R \ \, _2F_2\left(\frac{1}{2},1;\frac{3}{2},\frac{3}{2};-\frac{2 R^2}{\sigma ^2}\right)$$
confirmed numerically for $R=1, \sigma =1$