Post Made Community Wiki by Anton Geraschenko♦♦
1

[In front of a blackboard, in an office at Real College]

Skeptic: And why should I care about holomorphic functions?

Holomorphic enthusiast:$\;$ Can you compute $\quad$ $\sum_{n={-\infty}}^{\infty} \frac{1}{(a+n)^2}$ ? Here $a$ is one of your cherished real numbers, but not an integer.

Skeptic: Well, hm...

Holomorphic enthusiast, nonchalantly: Oh, you just get

$$\sum_{n={-\infty}}^{\infty} \frac{1}{(a+n)^2}=\pi^2 cosec^2 \pi a$$

It's easy using residues.

Skeptic: Well, maybe I should have a look at these "residues".

Holomorphic enthusiast (generously): Let me lend you this introduction to Complex Analysis by Remmert, this one by Lang and this oldie by Titchmarsh. As Hadamard said: "Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe".You can look for a translation at Mathoverflow. They have a nice list of mathematical quotations, following a question there.

Skeptic: Mathoverflow ??

Holomorphic enthusiast (looking a bit depressed) : I think we should have a nice long walk together now. [Exeunt]