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?

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    This is largely a matter of experience and pattern matching, not a specific and dedicated study to the art of obtaining closed form solutions. Two areas that are particularly conducive to finding these closed-form expressions are differential equations and probability. There are quite a few special functions that arise as families of solutions to particular ODE/PDE or as some explicit probabilistic expression. – Christopher A. Wong Jul 22 '14 at 19:40
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    @ChristopherA.Wong: Yes, I do agree that it is a matter of experience and pattern matching, but first you need to know how polylogarithm is defined and you should know some identities about it before you get experienced enough to see patterns. My question is where I should start from, not how I can be like these people in a week or a month! – math.n00b Jul 22 '14 at 19:59
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    @math.n00b my suggestion would be to learn identities in trig plus the integrals and power-series (if they exist) of the key "fundamental" functions (e.g., log, sin, cos, exponential, polynomials, and products thereof, plus some key rational functions like elliptic integrals). The more advanced users also rely on advanced tools from measure theory and abstract algebra (to name just two subjects I am familiar with). Above all else...**solve lots of math problems** requiring you to find closed form expressions. From my own experience, I never really understood something withouth solving problems –  Jul 22 '14 at 20:04
  • @Eupraxis1981: See here: http://math.stackexchange.com/questions/523027/a-math-contest-problem-int-01-ln-left1-frac-ln2x4-pi2-right-frac Do you think knowing trigonometric identities or elementary functions can help you in anyway in here? And I don't see how measure theory or abstract algebra relates to this. – math.n00b Jul 22 '14 at 20:13
  • @math.n00b The question referenced above is extremely difficult, and the answerer relied on a vast amount of experience *recognizing* different parts of the formula as having standard formulations. This is probably not a good prototype for you to be focusing on, as the vast majority of MSE folks could not solve this either. My advice is to read O.L.'s response and suggestions and then get yourself a lot of problems with *detailed* solutions and start working them. Math is like any skill...you only get good by practice...and start simple! –  Jul 23 '14 at 12:49

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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.

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    I respectfully disagree with your implication that such computational skill is a "waste of time" because such integrals are either already known or are computable with software. To the first point, we nevertheless teach basic calculus to students despite the existence of integral tables. Why? What function does this serve? To the second point, as a user of *Mathematica* since version 2.5, I am intimately familiar with the fact that there remain many integrals for which either evaluation in closed form is elusive, or at best is suboptimal in complexity. – heropup Jul 22 '14 at 23:15
  • Regarding the above: try evaluating the real-valued integral $$\int \frac{x^4 - 1}{(x^4 + 6x^2 + 1)\sqrt{x^4 + x^2 + 1}} \, dx$$ and try to get *Mathematica* to simplify it as much as possible. Do you think this is the simplest expression for an antiderivative? – heropup Jul 22 '14 at 23:23
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    @heropup Of course there are [many examples](http://en.wikipedia.org/wiki/Risch_algorithm#Problem_examples) of this kind. However one question is: why would one like to compute this particular integral? And another: ok assume we computed something - what have we learnt from this? How does this decrease the entropy in science? – Start wearing purple Jul 22 '14 at 23:30
  • So, what should be my first step to take? I should study complex analysis? Could you introduce a good complex analysis book that serves my purpose? – math.n00b Jul 23 '14 at 09:53
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    @math.n00b O.L. included a list of subjects above (bulleted list): introductory complex analysis, ODEs, elliptic functions. Plus, get yourself one of the books on computable integrals: Gradshteyn-Ryzhik or Prudnikov-Brychkov-Marychev. No one here is going to offer a detailed curriculum for you. Each person learns differently, but O.L.'s suggestions are good. –  Jul 23 '14 at 12:52
  • @Eupraxis1981: Gradshteyn-Ryzhik have gathered a huge table of integrals and series ;) It's not a book for learning, but rather a reference book for engineers. :) And you probably know that complex analysis is a really vast subject. Just saying introductory complex analysis is not enough. An introductory course to complex analysis might focus on topological theorems in complex analysis like open mapping theorem, maximum module principle, Schwartz lemma, Rouche's theorem and such results or it might focus on calculation of integrals and etc. – math.n00b Jul 23 '14 at 12:57
  • @math.n00b I wasn't suggesting reading a reference book cover to cover, just that "good" mathematicians will be able to recognize when one of those integrals come up, so you need to be familiar with them as you solve problems. As for the latter, just look on amazon for "introductory complex analysis" or look on MIT opencourseware to see what MIT considers a good book for intro complex analysis:http://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-fall-2003/syllabus/...if its good enough for MIT, you may want to look at it...seems to cover standard topics. –  Jul 23 '14 at 13:01
  • @math.n00b sorry, link didn't print right: http://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-fall-2003/syllabus/ –  Jul 23 '14 at 13:02
  • @Eupraxis1981: I have self studied most of those topics before, I haven't mastered them, but I know them relatively well. Yet, none of them seem related to the kind of problems I showed you. – math.n00b Jul 23 '14 at 13:07
  • @math.n00b you seem very focused on that particular problem. Again, most MSE folks wouldn't be able to solve that either, and there are a lot of well versed math folks on here. If you really want to solve THAT problem, then you need to look up what Li(x) is and the other concepts he mentioned. Each problem is unique, so you're not going to get good at getting "closed forms" as a general skill...its very context dependent, so the person who solved that integral may not be able to solve another integral...such abilities are rarely generalizable, as they rely on particular "tricks". –  Jul 23 '14 at 13:10
  • @math.n00b By saying "elementary complex analysis" I mean essentially residue theorem and manipulating multivalued functions. I cannot suggest a reference since I do not know well the literature in English, but any canonical textbook would be okay. The important part is solving exercises - you can find a lot of them, say, in Whittaker-Watson or even here on MSE. – Start wearing purple Jul 23 '14 at 13:11
  • @O.L.: You know, my main question is how a pure math student should learn about special functions on his own? I know that physicists have to deal with special functions a lot, but pure mathematicians don't learn about special functions during their undergraduate curriculum, at least not in my university. There is the same problem with vector calculus. Physicists learn a lot of things about vector/tensor analysis, while math students don't learn such stuff. For example, in one of your answers you refer to Hurwitz function. How did you encounter this function for the first time in your career? – math.n00b Jul 23 '14 at 13:18
  • @math.n00b I don't remember :) (about Hurwitz zeta) There are several mathematical routes that lead to special functions, most importantly linear differential equations and representation theory. In a sense, physicists take a shortcut via quantum mechanics which effectively (though often implicitly) deals with both subjects. – Start wearing purple Jul 23 '14 at 13:35
  • @O.L. And pure mathematicians will later learn about them when they study linear differential equations and representation theory? How about modular functions in number theory? Are they also a good source for these things? – math.n00b Jul 23 '14 at 13:40
  • @math.n00b There are different types of special functions. Linear ODEs and representation theory naturally lead to hypergeometric stuff, and I think for a long time the general feeling was that modular functions live in a completely different world. This, however, turned out to be wrong with the discovery of Monstrous Moonshine: you can take a look at [this answer](http://math.stackexchange.com/a/424865/73025) to get some vague idea. – Start wearing purple Jul 23 '14 at 13:52
  • @O.L. So, I shouldn't be worried about my math knowledge and think that I'm dumb because I don't understand such things yet, while undergrad physics students know and understand them, because later in my career I'll get the opportunity to study these things. Right? – math.n00b Jul 23 '14 at 14:01
  • @math.n00b Most probably yes. What I'm trying to emphasize is that special functions are (i.m.o.) worth studying not by themselves but only from some perspective, which could be the number theory, quantum mechanics, representation theory or something else. – Start wearing purple Jul 23 '14 at 14:16