From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: erf Date: 19 Jan 2000 02:14:39 -0500 Newsgroups: sci.math Summary: [missing] In article , Nicolas Bray wrote: : :Well, not quite. I know a couple standard proofs that: (Not quite what? I wish I could see the previous article, but this one looks like a start of a thread.) >Integral(e^(-x^2),x,-inf,inf)=sqrt(pi) > >but I'm guessing there are more out there. Anyone have some interesting >and/or obscure proofs of this? Depending what's standard in whose eyes. I can offer an identity involving a version of erf: int[0 to x] exp(-t^2) dt = sqrt(pi/4 - exp(-x^2) * A(x)) where A(x) = int[0 to 1] exp(-x^2*u^2)/(1+u^2) du and the classical result follows as you take x to infinity. Observe that A(x) is bounded; actually goes to 0 for x going to infinity. The identity is obtained by applying Fubini's Theorem to a suitable double integral. Another proof that seems "standard" to me because it was part of my undergraduate text: take integral[0 to sqrt(n)] (1 - x^2/n)^n dx (integral of a polynomial for every n), make a trigonometric substitution and end up with a product related to Wallis's product representing pi. By Lebesgue's Dominated Convergence Theorem, the integrals converge to the integral from 0 to infinity of exp(-x^2). Have fun, ZVK(Slavek).