Origin of identity: $\int\limits_{-\infty}^{\infty} \exp\{ - \pi x^2 - 2 \pi^{1/2} a x\} \,da = \exp\left\{a^2\right\}$

Question

A 1959 paper written by J. Hubbard called "Calculation of Partition Functions" and published in Physical Review Letters contains the following identity (Equation 2):

$$\int\limits_{-\infty}^{\infty} \exp\{ - \pi x^2 - 2 \pi^{1/2} a x\} \,dx = \exp\left\{a^2\right\}$$

Which can also be written as:

$$\frac{1}{\sqrt{\pi}} \int\limits_{-\infty}^{\infty} \exp\left\{2ax-x^2\right\} \,dx = \exp\left\{a^2\right\}$$

I am curious if anyone knows what the exact origin of this identity is, e.g., when was it first published and by whom? Hubbard does not provide any reference in his paper.

Answer 1

This is a variation on the Gaussian integral $\int\limits_{-\infty}^{\infty} \exp\left\{-x^2\right\} \,dx=\sqrt{\pi}$, a.k.a. Poisson or Euler-Poisson integral, to which it reduces by completing the square in the exponent:

$ \int\limits_{-\infty}^{\infty} \exp\left\{2ax-x^2\right\}\exp\left\{-a^2\right\} \,dx\\ = \int\limits_{-\infty}^{\infty} \exp\left\{-(x-a)^2\right\}\,dx=\sqrt{\pi}. $

De Moivre discovered it in 1733, Gauss published a proof in 1809, Poisson gave a neat trick for evaluating it using polar coordinates, popularized in Sturm's Cours d’Analyse de l’ecole polytechnique (1877).

But the first proof appears to be due to Laplace in Mémoire sur la probabilité des causes par les évènements (1774). See The probability integral for a list of other tricks by various authors with citations to the original sources.