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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    $\begingroup$ 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. $\endgroup$ – Christopher A. Wong Jul 22 '14 at 19:40
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    $\begingroup$ @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! $\endgroup$ – math.n00b Jul 22 '14 at 19:59
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    $\begingroup$ @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 $\endgroup$ – user76844 Jul 22 '14 at 20:04
  • $\begingroup$ @Eupraxis1981: See here: math.stackexchange.com/questions/523027/… 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. $\endgroup$ – math.n00b Jul 22 '14 at 20:13
  • $\begingroup$ @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! $\endgroup$ – user76844 Jul 23 '14 at 12:49

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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    $\begingroup$ 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. $\endgroup$ – heropup Jul 22 '14 at 23:15
  • $\begingroup$ 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? $\endgroup$ – heropup Jul 22 '14 at 23:23
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    $\begingroup$ @heropup Of course there are many 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? $\endgroup$ – Start wearing purple Jul 22 '14 at 23:30
  • $\begingroup$ 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? $\endgroup$ – math.n00b Jul 23 '14 at 9:53
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    $\begingroup$ @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. $\endgroup$ – user76844 Jul 23 '14 at 12:52

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