What is the naming reason of poles in complex analysis? - MathOverflow

What is the naming reason of poles in complex analysis?

2010-09-21T19:55:29Z

A function $f: \textbf{C} \to \textbf{C}$ has a pole of order $k$ if $f(z) = \frac{g(z)}{(z-z_0)^{k}}$ where $g(z)$ is a nonzero analytic function. Why do we call it poles?

Answer by anon for What is the naming reason of poles in complex analysis?

2010-09-21T23:00:00Z

This may be apocryphal folk etymology, but I always thought it was because if you plot, or envision plotting, the surface z = |f(x + iy)|, at poles of f, the surface, if you imagine it sitting over the xy plane, looks like it is being supported by a really tall pole. Like a circus tent. I have no citations to support this belief, but I must have gotten it from somewhere. Anyway it makes a good deal of sense.

I'm posting this, despite not having an MO account, because genuinely can't understand why nobody has posted it yet. (Nikita's "because poles stick up" comes close, but seems to have been drowned out by posts about poles being "big", or invocations of the north pole, which seem to be entirely different explanations.)

Answer by Richard Borcherds for What is the naming reason of poles in complex analysis?

2010-09-22T02:32:09Z

This picture http://en.wikipedia.org/wiki/File:Jahnke_gamma_function.png should make it clear why they are called poles.

Answer by Greg Graviton for What is the naming reason of poles in complex analysis?

2010-09-22T09:36:22Z

As Martin O remarks, the french word "pôle" (or the German "Pol") does not include the meaning "pillar", that's why the explanation "looks like a pillar" seems a bit dubious, at least to me. Being a native German speaker, I have always associated the word "pole" to the electrical pole and thus to the function $\frac1{r^2}$. It seems entirely natural to me to use this special example as a prototype for any singularity of a (complex) function.

Answer by Andrey Rekalo for What is the naming reason of poles in complex analysis?

2010-11-07T04:35:50Z

According to these pages, the earliest known appearance of the term pole might be in "Théorie des fonctions elliptiques" (1875, p. 15) by Briot and Bouquet:

Lorsqu'une fonction $u$ est holomorphe dans une certaine partie du plan, excepté en un point $z_1$, où elle devient infinie, de manière toutefois que la fonction $\frac{1}{u}$ reste holomorphe dans le voisinage de ce point, on dit que ce point est un pôle ou un infini de la fonction $u$.

They don't provide any motivation for this choice of a term though.

By the way, in their first memoir on the subject, "Étude des fonctions d'une variable imaginaire" (1856), Briot and Bouquet refer to a pole of a function only as un infini du degré fini.