Probability and Laplace/Fourier transforms to solve limits/integrals from calculus.

Question

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.

Answer 1

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.