What are your favorite relations between e and pi? [closed]

This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for mathematics due to some personal stuff in his life. (I will NOT discuss his personal life, but it has nothing to do with me).

To make him feel better about math (or love of math for that matter), I was planning on giving him a sheet of paper with quite a few relations of $e$ and $\pi$

The ones I were going to give him were:

$$e^{i\pi}=-1$$

and how $e$ and $\pi$ both occur in the distribution of primes (bounds on primes has to do with $\ln$ and the regularization of the product of primes are $4\pi^2$)

Can I have a few more examples of any relations please? I feel it could mean a lot to him. I'm sorry if this is too soft for this site, but I didn't quite know where to ask it.

I'm a fan of $$e^\pi-\pi=20{}$$(Well... almost...)

$$\int_{-\infty}^\infty\frac{\cos x}{x^2+1}\operatorname d\!x=\frac\pi e$$ EDIT: Also: $$\int_{-\infty}^\infty e^{-x^2}\operatorname d\!x=\sqrt\pi$$

Some additional possibilities

$$n!\sim\sqrt{2\pi n}\left(\frac {n}{e}\right)^n$$

The normal distribution is given by

$$\phi(x) = \frac{1}{2 \pi}e^{(-1/2)x^2}$$

$$\int_{-\infty}^\infty\phi(x)dx=1$$

A personal favorite involving the Euler–Mascheroni constant.

$$\int_0^\infty e^{-x}\ln^2x \,dx=\gamma^2+\frac{\pi^2}{6}$$

Also

$$\frac{e^{3+2\gamma}}{2\pi}=\prod_{n=1}^\infty e^{-2+2/n}\left(1+\frac{2}{n}\right)^n$$

Some rather unusual ones

$$\sum_{n=1}^\infty \frac 1 {n^2}\cos\left(\frac 9{n\pi+\sqrt{n^2\pi^2-9}}\right) =-\frac{\pi^2}{12e^3}$$

$$\pi=72\sum_{n=1}^\infty \frac{1}{n(e^{n\pi}-1)}-96\sum_{n=1}^\infty \frac{1}{n(e^{2n\pi}-1)}+24\sum_{n=1}^\infty \frac 1 {n(e^{4n\pi}-1)}$$

Edit:

$$\zeta(3)=\frac{7}{180}\pi^3-2\sum_{k=1}^\infty \frac 1 {k^3(e^{2\pi k}-1)}$$

An approximation for the Feigenbaum constant is given by

$$\pi+\tan^{-1}\left(e^\pi\right)$$

$$\int_0^\infty \Gamma(x)\,dx=e+\int_0^\infty\frac{e^{-x}}{\pi^2+\ln^2x}\,dx$$

$$\prod_{k=0}^\infty \frac{1}{k^{1/k^2}}=\left(\frac{A^{12}}{2\pi e^\gamma}\right)^{\pi^2/6}$$

where $A$ is the Glaisher–Kinkelin constant.

The denominators $q_n$ of the convergents of the continued fraction expansions of almost all real numbers satisfy

$$\lim_{n\to\infty}q_n^{1/n}=e^{\pi^2/(12\ln 2)}$$

This is known as Lévy's constant.

Here's a beautiful relation between $\pi$ and $e$,

$$\sqrt{\frac{\pi\,e}{2}}=1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots+\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}}$$

It shouldn't be hard to guess who found this. As Kevin Brown of Mathpages remarked, "Is there any other mathematician whose work is instantly recognizable?"

I honestly just like the fact that $e + \pi$ might be rational. This is the most embarrassing unsolved problem in mathematics in my opinion. It's clearly transcendental and we have no idea how to prove that it's even irrational.

They're so unrelated additively that we can't prove anything about how unrelated they are. Uh-huh.

$e\pi$ might also be rational. But we do know they can't both be (easy proof).