How to calculate integrals like $\int_{-\infty}^{\infty} e^{-x^2} dx$?

Question

My question as hinted at in the headline, do you have a recommendations for a book, where explicitly these types of problems are discussed? No sure about $\int_{-\infty}^{\infty} ...dx$.

What is the level of math you know, that might help us in suggesting you references. — user35952, Feb 06 '23 at 12:47
I am an undergrad. But this should not be a concern. I am willing to to dive deeply into the matter. maybe a book which is not so technical and a book which goes really into depth. — mathsdummie1234, Feb 06 '23 at 12:49
Here there is the proof: https://jakubmarian.com/integral-of-exp-x2-from-minus-infinity-to-infinity/ — Sebastiano, Feb 06 '23 at 12:58
Check this page for resource reccommendation on books - https://math.stackexchange.com/q/176032/116937
A very introduction would be Schaum's series (Advanced calculus/Outline of Calculus) — user35952, Feb 06 '23 at 13:05
Since you tagged [complex-analysis], see this video explaining a method to compute it via contour integral — user170231, Feb 06 '23 at 16:05
You'd probably like 'Advanced Calculus Explored' by H. Alsamraee — Sine of the Time, Feb 06 '23 at 16:41

score 2 · Answer 1 · answered Feb 06 '23 at 13:02

2

This integral is very famous and is called the Gaussian integral. There is a wikipedia page dedicated to this integral where you can find multiple proofs that $$ \int_{-\infty}^{+\infty}e^{-x^2}dx=\sqrt{\pi}. $$ As for studying integrals that involve infinite bounds (they are called improper integrals) you can read the great wikipedia page.

answered Feb 06 '23 at 13:02

Tuvasbien

8,907

Thanks, thats clarifying all. The english version of wiki is so much better than other languages. – mathsdummie1234 Feb 06 '23 at 13:04

How to calculate integrals like $\int_{-\infty}^{\infty} e^{-x^2} dx$?

1 Answers1