Posted 21 July 2007 - 04:21 AM
It is of an interesting note that in general the vast majority of functions cannot be easily integrated into a closed form, by this, I mean NOT a power series expansion, which is how we define the "imaginary error function" as noted by lambus. As noted by Sanctus, it is certainly possible to integrate this function over the entire real line, or any infinite interval. Interestingly enough, you can integrate a function of the form f(x) = x^5 * e^(x^2). Before ripping your hair out, make a rationalizing substiution and don't be afraid to do make another substiution and apply parts more than once. It's a fun integral, and interesting, I might say. I have actually meant to take time to look at functions of the form f(x) = x^m * e^(x^k), but I have yet to get around to that.
And the answer to your question is a resouding no, no we cannot integrate any function. The function e^(x^2) is of a class a functions who antiderivatives are defined as transcendental (neither elementary nor algebraic) and are not expressed in terms of a "normal" function. And the reason for there be no explicit formulae (as opposed to an infinite chain of polynomials) for certain integrals is rather complicated one, and I think I need some time to brood over that one (when it's not six in the morning). But interesting question, you'd never believe how many people get their doctorates in mathematics and never ask such simple questions.
And Sanctus, I've seen the derivation of this integral, but the one thing I never understood is why the integral squared is equal to the integral times the integral with respect to y. This is probably due to my unfamiliarity with multivariable calculus.