How to find the integral $\int_{-\infty}^{\infty}\frac{dx}{1+ae^{bx^2}}$

Question

Could somebody tell me how to find the integral $$\int_{-\infty}^{\infty}\frac{dx}{1+ae^{bx^2}}$$ for constants $a$ and $b$?

Thanks!

Answer 1

I will assume that $a, b$ are positive constants, in order to circumvent singularity issues. By the substitution $z = \sqrt{b} \, x$, the integral in question is equal to

$$ \frac{1}{\sqrt{b}} \int_{-\infty}^{\infty} \frac{e^{-z^2}}{a + e^{-z^2}} \; dz.$$

From the identity

$$ 1 + x^{2n+1} = (1 + x)(1 - x + \cdots - x^{2n-1} + x^{2n}), $$

we obtain

$$ \frac{1}{1 + x} = 1 - x + \cdots - x^{2n-1} + x^{2n} - \frac{x^{2n+1}}{1 + x}.$$

Now we temporary assume further that $a > 1$, so that $\alpha = a^{-1} \in (0, 1)$. Then

$$\begin{align*} \int_{-\infty}^{\infty} \frac{e^{-z^2}}{1 + \alpha e^{-z^2}} \; dz &= \int_{-\infty}^{\infty} \left( \sum_{k=1}^{2n+1} \alpha^{k-1} e^{-kz^2} - \frac{\alpha^{2n+1}e^{-(2n+2)z^2}}{1 + e^{-z^2}} \right) \; dz \\ &= \sum_{k=1}^{2n+1} (-1)^{k-1} \alpha^{k-1} \int_{-\infty}^{\infty} e^{-kz^2} \, dz - \alpha^{2n+1} \int_{-\infty}^{\infty} \frac{e^{-(2n+2)z^2}}{1 + e^{-z^2}} \; dz \\ &= \sum_{k=1}^{2n+1} (-1)^{k-1} \alpha^{k-1} \sqrt{\frac{\pi}{k}} - \alpha^{2n+1} \int_{-\infty}^{\infty} \frac{e^{-(2n+2)z^2}}{1 + e^{-z^2}} \; dz \end{align*}$$

Now taking $n\to\infty$, the remainder term vanishes. Hence we have

$$ \int_{-\infty}^{\infty} \frac{e^{-z^2}}{1 + \alpha e^{-z^2}} \; dz = \sum_{k=1}^{\infty} (-1)^{k-1} \alpha^{k-1} \sqrt{\frac{\pi}{k}} = -a \sqrt{\pi} \, \mathrm{Li}_{1/2} \left( -\tfrac{1}{a}\right),$$

where

$$ \mathrm{Li}_{s}(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^s}$$

is the polylogarithm of order $s$, primarily defined on $|z| < 1$. Although we have proved this identity only for $a > 1$, the equality above can be used to define an analytic continuation of the right hand side, thus (by tautology) it holds for all $a > 0$.

It has special value at $\alpha = 1$, given by

$$ \int_{-\infty}^{\infty} \frac{1}{1 + e^{z^2}} \; dz = -\sqrt{\pi} \, \mathrm{Li}_{1/2}(-1) = \sqrt{\pi} (1 - \sqrt{2}) \zeta \left( \tfrac{1}{2} \right)$$