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I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let $X$ be a random variable from $[0,1]$... some examples are in those answers, see also this for an example that has to do with evaluation of limit of a series) or some uses Laplace transforms or even Fourier series (Example see some answers by user Tunk-Fey: 1, 2 ). I would like to ask about a book that discusses those techniques, i.e. where to learn them?

$\bullet$ Edit: There's no problem if those techniques can only be learned from a particular section of one book or more (or even just a paper, as long as it is accessible to an upper-undergrad.).

Thanks in advance, that would really help me unlock my box of tools.

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    $\begingroup$ The Laplace transform and Fourier series will be discussed in pretty much any differential equations textbook. $\endgroup$ – Chantry Cargill Dec 24 '14 at 20:54
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    $\begingroup$ I guess "Inside Interesting Integrals: A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, " and Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals will be a good place to start searching $\endgroup$ – SomeOne Dec 24 '14 at 20:55
  • $\begingroup$ @Ahmed I have both books, but none contains a treatment of probabilistic methods. However Nahin's book contains a small discussion of Fourier series and how they can be used to determine the value of an integral using one example. $\endgroup$ – user153330 Dec 24 '14 at 21:08
  • $\begingroup$ @ChantryCargill My focus is not on the theory behind Laplace and Fourier transforms but on how they can be used to find integrals. $\endgroup$ – user153330 Dec 24 '14 at 21:13
  • $\begingroup$ You might also check out the artofproblemsolving forum. I'm sure they have tons of arguments using the methods you've mentioned. $\endgroup$ – Robert Cardona Dec 24 '14 at 22:08
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In my opinion, the best books to learn these stuff may be found in physics and engineering textbooks. Here are some reference books:

  1. Introduction to Methods of Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers by Sean Mauch
  2. Mathematical Methods for Physicists: A Concise Introduction by Tai L. Chow
  3. Mathematics for Electrical Engineering and Computing by Mary P Attenborough
  4. Mathematics for Physics I by Michael Stone and Paul Goldbart
  5. Mathematical Methods in the Physical Sciences by Mary L. Boas

My most personal recommendation book to learn these three subjects is book number 5. In addition, you might find these old books interesting to learn (although, these books don't discuss those subjects specifically): A Treatise on Integral Calculus Volume 1 and Volume 2 by Joseph Edwards. Well, I hope all these books can help you.

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  • $\begingroup$ I've given you the bounty since it would have been gone, anyway the question is still unanswered but thanks for the references, I didn't know them : ) $\endgroup$ – user153330 Dec 29 '14 at 20:35
  • $\begingroup$ @user153330 Thank you but what kind of answer do you want? I'll add some details to my answer. $\endgroup$ – Venus Dec 30 '14 at 5:03

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