Classes

distler Administator 20 posts

edited 12 years ago

In conversation at Office Hours, it came up that not every integral can be written in terms of elementary functions.

In particular, there’s a function called the “Error Function,”

Erf (s) = \frac{2}{\sqrt{π}} \int_{0}^{s} e^{- t^{2}} d t

Of course, we have that $Erf (s)$ is a monotonic function, with $Erf (0) = 0$ and (more nontrivially) $Erf (∞) = 1$ .

To prove the latter, note that

\begin{aligned} {(\int_{0}^{∞} e^{- u^{2}} du)}^{2} & = \int_{0}^{∞} \int_{0}^{∞} e^{- (u^{2} + v^{2})} d u d v \\ = \int_{0}^{∞} e^{- r^{2}} r d r \int_{0}^{π / 2} d θ \\ = \frac{π}{4} \int_{0}^{∞} e^{- x} d x \\ = \frac{π}{4} \end{aligned}

Hope that helps …

Integral