Who first found the value of $\int_{-\infty}^{+\infty}e^{-x^2}dx$?

Question

A fairly pretty technique of showing that
$$\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt{\pi}$$ is to square the integral, writing that square as the product of two integrals with integration variables $x$, and $y$, treating that as an integral over the whole plane $\Bbb{R}^2$, and then changing to polar coordinates, where the resulting integrand picks up a factor of $r$ to become easy to integrate.

Clearly, this is an important integral when discussing the normal, or Gaussian, distribution. And I had always assumed that Gauss was the first to determine this integral. But I'm thinking that maybe somebody else derived this result before Gauss did. Does anybody know who got it first?

Answer 1

IIRC the textbook we used for the Analysis II course in Belgrade 25 years ago (roughly corresponding to honour Calc III and non-existing Calc IV in U.S.) refereed to that integral and technique you described as Poisson integral. Siméon Denis Poisson and Carl Friedrich Gauss are contemporaries as you probably already know but I really like the answers posted to related question .

Answer 2

In the 1720s, de Moivre had discovered the normal approximation to the binomial distribution. In the course of this work de Moivre showed $\binom{2n}{n}/2^{2n} \sim C/\sqrt{n}$ for some $C$ that he could estimate numerically using infinite series. Stirling identified $C$ as $1/\sqrt{\pi}$ (which is equivalent to Stirling's formula for $n!$) but he did not prove this. Figuring out $C$ is equivalent to figuring out the Gaussian integral in the question, so one could say Stirling found the value of the Gaussian integral, but he did not derive it in any systematic way. While de Moivre was able to prove $C = 1/\sqrt{\pi}$ in 1730 using the Wallis product for $\pi$, I do not think he viewed it as equivalent to a calculation of $\int_{-\infty}^{\infty} e^{-x^2}\,dx$.

The first proof that the integral in the question is $\sqrt{\pi}$, based on working directly with integrals, is due to Laplace in 1774 in his paper on inverse probability. He used a formula of Euler to show $$ \int_0^1 \frac{dx}{\sqrt{-\log x}} = \sqrt{\pi}. $$ Under the change of variables $y = \sqrt{-\log x}$ the integral becomes $2\int_0^{\infty} e^{-y^2}\,dy$, which is $\int_{-\infty}^{\infty} e^{-y^2}\,dy$.

Laplace later (1812) gave a second proof based on squaring the Gaussian integral and making a change of variables, but it was not based on polar coordinates. The use of polar coordinates is due to Poisson. See http://www.york.ac.uk/depts/maths/histstat/normal_history.pdf for a survey of early proofs.