reference request - How do people on MSE find closed-form expressions for integrals, infinite products, etc?

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I've seen some pretty complicated integrals and infinite products and infinite series and other math equations that have been reduced to simple closed form expressions using special functions like polylogarithms, Riemann's zeta function, Bessel functions and other such functions.

People like Vladimir Reshetnikov, Robjohn, Eric Naslund, SOS440 among many others have always impressed me with their ability to find closed form expressions for very complicated mathematical formulas.

The problem is that this part of mathematics, which I find it incredibly beautiful and aesthetic, is usually not covered in a typical math curriculum. I tried to find a book that explains how such closed form expressions are found with lots of problems to practice but I haven't found any yet. I checked some books about hypergeometric series, but they were rather too technical and they preferred to focus on analysis of hypergeometric series, not their manipulations and techniques of doing calculations with them.

My main question is that if I want to be nearly as good as the people I mentioned in finding closed form expressions, where should I start from? How much math background and maturity do I need? Can you suggest any good books or video lectures for that?

Answer

I would not call this "part of mathematics" beautiful nor aesthetic, although sometimes it can be a pleasant waste of time. Most of the computable integrals, sums and products can be found in the books like Gradshteyn-Ryzhik or Prudnikov-Brychkov-Marychev. Programs like Mathematica or Maple, as well as theoretical physicists, solve this kind of problems already quite efficiently.

There is a finite number of patterns/recipes/tricks to apply if you want to do this yourself. This comes with practice. The most important piece for understanding is elementary complex analysis. Two natural further directions are:

• a bit of linear ODEs in the complex domain. This helps to understand the origin of various properties of special functions of hypergeometric type without remembering any of them.




• elliptic functions. In addition to being useful in computations, this actually is a very beautiful and deep piece of mathematics. Especially if you don't stop at Whittaker-Watson but continue, say, with Mumford lectures.