Gaussian Summability

Lemma 1. (Complex Gaussian Integral) Suppose $s \in \mathbb{C}$ satisfies $s \neq 0$ and $\Re (s) \geq 0$. Then \[ \int_{-\infty}^{\infty} e^{-s x^2} \, dx = \sqrt{\frac{\pi}{s}}, \] where $\sqrt{z} = \exp(\frac{1}{2} \text{Log} z)$ denotes the principal square root of $z \in \mathbb{C} - (-\infty, 0)$.

Pf. For $\Re (s) > 0$, the change of variable with $x = \sqrt{y}$ yields \[ \int_{-\infty}^{\infty} e^{-s x^2} \, dx \ = \ 2 \int_{0}^{\infty} e^{-s x^2} \, dx \ = \ \int_{0}^{\infty} y^{-1/2} e^{-s y} \, dy. \] Then we know from basic theory of Laplace transform that this is analytic for $\Re (s) > 0$. But for positive real $s$, this coincides with the holomorphic function \[  \frac{\Gamma \left( \frac{1}{2} \right)}{\sqrt{s}} \ = \ \sqrt{\frac{\pi}{s}}, \] so they must conicide in $\Re (s) > 0$ as well. So it remains the case $\Re (s) = 0$ and $s \neq 0$. Note that it suffices to show that \[ \int_{-\infty}^{\infty} e^{- i x^2} \, dx = e^{- i \pi / 4} \sqrt{\pi}, \] which is clear from a simple application of contour integration along the wedge contour.

Theorem 2. Let $s \in \mathbb{C}$ satisfy $s \neq 0$ and $\Re (s) \geq 0$. If $\beta \in \mathbb{R}$, then we have \[ \int_{0}^{\infty} \exp \left( - s \left( x^2 + \frac{\beta^2}{x^2} \right) \right) \, dx \ = \ \frac{1}{2} \sqrt{\frac{\pi}{s}} e^{-2 |\beta| s}. \]

Pf. If $\beta = 0$, this reduces to the Lemma 1. So we may assume $\beta > 0$. Let's deonte the integral in question $I(\beta)$. Then the substitution $x \mapsto \beta / x$ yields \[ I(\beta) \ = \ \int_{0}^{\infty} \frac{\beta}{x^2} \exp \left( - s \left( x^2 + \frac{\beta^2}{x^2} \right) \right) \, dx. \] Then \[ I(\beta) \ = \ \frac{1}{2}I(\beta) + \frac{1}{2}I(\beta) \ = \ \frac{1}{2} \int_{0}^{\infty} \left( 1 + \frac{\beta}{x^2} \right) \exp \left( - s \left( x^2 + \frac{\beta^2}{x^2} \right) \right) \, dx. \] Finally, put $u = x - \beta / x$. Then this becomes \[ I(\beta) = \frac{1}{2} e^{-2s\beta} \int_{-\infty}^{\infty} e^{- s u^2} \, du, \] from which we deduce the desired identity.