# Tagged Questions

Use this tag for the error and complementary error functions (erf and erfc). These are special functions formed by taking definite integrals of the Gaussian/normal distribution function.

5k views

### What is the antiderivative of $e^{-x^2}$

I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space \text{erf} (x) + C$$ So, I ...
835 views

### How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?

How do I prove the error function $$\mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt.$$ is entire? Could you give me some scratch proof?
226 views

### Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
### How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$?
I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ \mathrm dx=\frac1{\sqrt\pi},\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ \mathrm dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2,$$ ...