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

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.

Thanks alot!!

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