# Prove $\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$

I came across this nice identity: $$\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$$

Is there an elementary proof?

• $\pi,\Gamma,e,\ln,\infty,0$ together. I am not sure if this is elegant or an abomination.
– Guy
Apr 5, 2014 at 16:31
• This is the Fransen-Robinson constant, the continuous equivalent of e, and whose approximate value is about $2.708880^+$. Apr 5, 2014 at 16:36
• IMHO, in a whole year I joined MSE, this is probably the most beautiful identity I have seen. Apr 5, 2014 at 16:37
• @Lucian (+1 for the link), what do you mean "continuous equivalent" of $e$ , are you referring to $$e=\sum_{n=0}^\infty \frac{1}{n!}?$$
– Guy
Apr 5, 2014 at 16:45
• @yzhao Why does that surprise me? Apr 5, 2014 at 17:24

Hardy recorded in his 1937 papers that the formula was discovered by Ramanujan, who did not profess to prove such an identity. Hardy published his proof, which was based on the "Plana's formula"(A proof of Plana's formula can be found here). Here is an outline of Hardy's original proof. I doubt whether an "elementary" proof exists(without complex analysis).

The Plana's formula asserts that

$$\sum_{n=0}^{\infty}f(n)-\int_{0}^{\infty}f(x)dx=\frac{1}{2}f(0)+i\int_{0}^{\infty}\frac{f(it)-f(-it)}{e^{2\pi t}-1}dt$$

Let $f(u)=1/\Gamma(u)$, then $$\sum_{n=0}^{\infty}\frac{1}{n!}-\int_{0}^{\infty}\frac{1}{\Gamma(x)}dx=i\int_{0}^{\infty}\frac{1/\Gamma(it)-1/\Gamma(-it)}{e^{2\pi t}-1}dt$$

An elementary transform of RHS gives

$$i\int_{0}^{\infty}\frac{1/\Gamma(it)-1/\Gamma(-it)}{e^{2\pi t}-1}dt=-\frac{1}{2}\int_{-\infty}^{\infty}\frac{1/\Gamma(it)}{\sin \pi i t}e^{-\pi\vert t\vert}dt$$

Also $$e^{-\pi\vert t\vert}=\int_{0}^{\infty}\frac{u^{it}}{u(\pi^2+\log^2u)}du$$

Then RHS is equal to $$\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{u^{it}}{u(\pi^2+\log^2u)}\frac{-1/\Gamma(it)}{2\sin \pi i t}dudt$$

Mellin transform gives $$-\frac{1}{2\pi i}\int_{C}\frac{\pi u^{z-1}}{\Gamma(z)\sin \pi z}dz=-e^{-u}$$

where C is the imaginary axis.

And we are done.

• +1. Nice solution. I was trying to use Euler Reflection Formula but it became a mess !!!. Apr 6, 2014 at 4:23