# Some users are mind bogglingly skilled at integration. How did they get there?

Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this.

Especially in a world where most scientists compute their integrals numerically, I find it astounding that people exist that can solve these problems analytically. Sometimes it is a truly bizarre contour, some examples use auxiliary functions and differentiation under the integral sign in a way that feels more like integration by wizardry.

Some of these users seem to have mastered an incredibly large number of special functions to a level of fluency that is almost unimaginable to me. Manipulations involving the error function almost looks like the work of an infant compared to some of these seemingly casual manipulations involving the Airy function, Barnes G function, the Legendre chi function and many more. Every time I read a post by them, it turns into an exercise in reading about yet another special function I have never heard of.

My question is what sort of material do you have to study, or what kind of area do you have to work in to get so good? To me, it doesn’t seem as obvious as saying something like, “oh, they know this material because they are a topologist”.

I hesitate to provide an answer to this because it feels very immodest, but perhaps I can provide something useful that doesn’t merely restate the obvious or succumb to narcissism on my part.

First of all, I will state something kind of obvious, as @FireGarden has pointed out. How do you get to Carnegie Hall? Practice! The people who I consider to be integration gurus on this site (@sos440, @O.L, @RandomVariable, @robjohn, @achillehui, forgive me if I left anyone out) do not strike me as Ramanujan-like savants who get their inspiration from the goddess Namagiri. Rather, these are people who clearly 1) love working out integrals (and sums and related quantities), and 2) work out an enormous number of problems, completely and in great detail. So, yes, hard work and enthusiasm explain a lot.

This doesn’t mean, however, that you sit with a textbook and merely work out the exercises in a boring, rote manner. What I mean by practice is the application of your knowledge to new, interesting problems. Find problems that look impossible to you right now. Maybe they are, but I guarantee you somebody somewhere has solved it. Here are some places to find interesting/challenging problems:

• Putnam exam problem archives (start here); keep in mind that several Putnam exam problems are solved in Math.SE.
• Actuarial P exam (tons of double integrals over interesting regions, as well as some interesting single integrals in the context of probability computations, here’s a sample)
• Graduate Physics texts (source of many integrals found here)

But practice is not the only thing. The really great aspect of a forum like SE is that we all learn from each other. So, as you look at these terrific solutions, don’t worry about how the author came up with the idea. Worry instead about learning from the insight, and see if you can apply the lessons elsewhere. I for one can say that I have totally benefitted from reading through the solutions of my peers here. Nobody here is an island.

Do not be intimidated by special functions; you will learn about them as you need them. Special functions really are just high-falutin’ definitions that one person or another has found useful at some time. They are only useful when they can be computed in far less operations than a numerical evaluation of the integral defining them. (I’ve ranted about this before.) They do not imply a level of mastery, but rather a level of familiarity with the art. Personally, I have no issue with answers expressed in terms of special functions as long as they represent the simplest closed form of a solution. Occasionally, one find poseurs who use fancy special functions when a much simpler form does the trick; usually they are exposed.

So what might you want to learn? In decreasing order of importance: 1) Calc II integration techniques (crucial to master), 2) Mathematica/Maple (for algebraic and numeric verification), 3) Ordinary differential equations (for exposure to basic special functions and series solution techniques), 4) Complex analysis (because the residue theorem is incredibly useful), 5) Integral transforms (Laplace, Fourier), although more for the cool integrals to do rather than any “techniques” taught, 6) Laplace’s Method, Method of Steepest Descent, Watson’s Lemma, and like asymptotic methods, 7) Statistical methods, especially the various PDFs used in various contexts. In some cases, knowledge of Lebesgue integration may be useful (although I plead guilty to inexperience there).

I hope this helps. I’ve likely left stuff out, so I may have more to say.