Finding representations of antiderivatives without a closed form

[Manchot]

I was wondering if anyone knew of any good books (or textbooks, or websites) which discuss finding series representations of integrals which exist, but don't have a closed form. I'm interested in the subject at the moment, but I haven't had much luck online. Furthermore, what branch of calculus would this fall under? I'd imagine that it would be some kind of analysis, but seeing as I've never taken an analysis course, I have no idea what type. (I've only taken Calculus I, II, III and Diff Eq). Thanks for the help!

EDIT: I forgot to say that I'm really only interested in the integrals of real functions. I guess that that's important.

 

[Crosson]

Obviously, any analytic function can be represented by a power series which can always be integrated term by term within an interval of convergence.

There are various special methods, for example the asymptotic series expansion for the error function.

Learn about special functions (I think the best way to do this is with mathematica), which varioous integrands cani be reduced to.

 

[dextercioby]

Either if you do learn about special functions (the most important i would consider to be the elliptical ones and the hypergeometric),u still may encounter integrals whose values cannot be expressed by neither of the functions "Mathematica" knows...

Daniel.

 

[Manchot]

As for power series, I'm not terribly interested in those, either. However, asymptotic series are one of my interests. I just now looked at the derivation for the asymptotic expansion for the error function, and was kind of disappointed, because it just used the "infinite integration by parts" method which I already figured out how to do. Are there any other such methods?

Yeah, I have been learning some of those special functions recently, but I'm curious as to how I should go about learning the elliptic integrals and hypergeometric functions. I just MathWorld'ed them and was kind of stunned by their complexity. Where should I even begin?