What is the naming reason of poles in complex analysis? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:08:30Z http://mathoverflow.net/feeds/question/39538 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39538/what-is-the-naming-reason-of-poles-in-complex-analysis What is the naming reason of poles in complex analysis? Trevor C 2010-09-21T19:55:29Z 2010-11-07T04:48:06Z <p>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? </p> http://mathoverflow.net/questions/39538/what-is-the-naming-reason-of-poles-in-complex-analysis/39547#39547 Answer by anon for What is the naming reason of poles in complex analysis? anon 2010-09-21T23:00:00Z 2010-09-21T23:00:00Z <p>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. </p> <p>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.)</p> http://mathoverflow.net/questions/39538/what-is-the-naming-reason-of-poles-in-complex-analysis/39574#39574 Answer by Richard Borcherds for What is the naming reason of poles in complex analysis? Richard Borcherds 2010-09-22T02:32:09Z 2010-09-22T04:19:23Z <p>This picture <a href="http://en.wikipedia.org/wiki/File:Jahnke_gamma_function.png" rel="nofollow">http://en.wikipedia.org/wiki/File:Jahnke_gamma_function.png</a> should make it clear why they are called poles. </p> http://mathoverflow.net/questions/39538/what-is-the-naming-reason-of-poles-in-complex-analysis/39589#39589 Answer by Greg Graviton for What is the naming reason of poles in complex analysis? Greg Graviton 2010-09-22T09:36:22Z 2010-09-22T09:36:22Z <p>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 <em>electrical pole</em> 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.</p> http://mathoverflow.net/questions/39538/what-is-the-naming-reason-of-poles-in-complex-analysis/45137#45137 Answer by Andrey Rekalo for What is the naming reason of poles in complex analysis? Andrey Rekalo 2010-11-07T04:35:50Z 2010-11-07T04:48:06Z <p>According to these <a href="http://jeff560.tripod.com/mathword.html" rel="nofollow">pages</a>, the earliest known appearance of the term pole might be in <a href="http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&amp;O=NUMM-99571" rel="nofollow"><em>"Théorie des fonctions elliptiques"</em></a> (1875, p. 15) by Briot and Bouquet: </p> <blockquote> <p>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 <em>pôle</em> ou un <em>infini</em> de la fonction $u$.</p> </blockquote> <p>They don't provide any motivation for this choice of a term though.</p> <p>By the way, in their first memoir on the subject, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k995680.r=briot.langEN" rel="nofollow"><em>"Étude des fonctions d'une variable imaginaire"</em></a> (1856), Briot and Bouquet refer to a pole of a function only as <em>un infini du degré fini</em>. </p>