# What is the naming reason of poles in complex analysis?

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?

• It's because they correspond to the north pole when considered as holomorphic functions onto the riemann sphere. – muad Sep 21 '10 at 20:50
• Insert your favorite Polish joke here. – Nate Eldredge Sep 21 '10 at 21:21
• I, too, would like this question stay open. Please, count it as a vote against closing per old meta discussion. – Victor Protsak Sep 21 '10 at 22:21
• @Nate: There was a transatlantic flight and the pilot and copilot dropped dead. A desperate flight attendant asked if anyone knew how to fly a plane. An old polish man said "Well, I used to fly planes in WWII, but nothing like this". When she brought him into the cockpit, his jaw dropped. There was so many buttons, levers, and fancy dials. "What's wrong?" the flight attendant asked. "I'm just a simple pole in a complex plane" he responded. – David Carchedi Sep 21 '10 at 23:20
• I hope this comment counts as a vote against closing too. The question seems to be a hard one, well worth the attention of the MO community. I notice that the historian Judith Grabiner asked this question on an internet forum in 1998, and she apparently did not receive an answer. – John Stillwell Sep 22 '10 at 0:10

The answer suggested by @muad’s comment is supported by both P. Ullrich (1989, p. 163):

The expressions “pole” and “polar singularity” were apparently first used in the 1865 book “Vorlesungen über Riemann’s Theorie der Abel’schen Integrale” of Carl Neumann (1832–1925): they are precisely the points that the function maps to the “pole” of the Riemann sphere. Neumann’s book, by the way, is also the first to develop the theory of the Riemann sphere, which Riemann himself had only presented in his lectures. (See Neuenschwander (1978) (...).)

(my emphasis) and A. Markushevich (1996, p. 191):

The classification of the singularities of an algebraic function as poles and critical points was carried out, for example, in the Théorie des fonctions elliptiques of Briot and Bouquet (1875). However, the term “pole,” as pointed out by E. Neuenschwander (1978) was first used in this sense by K. Neumann in his Vorlesungen über Riemann’s Theorie der Abelschen Integrale (1865) in connection with the fact that the point at infinity was depicted as the pole of the sphere in this book.

The quoted article of Neuenschwander notes that the histories by Brill-Noether (1894, p. 170) and Osgood in the Encyklopädie (1901, p. 18) both miscredit Briot and Bouquet. So does (as of now) the earliest uses site — quoting an edition (1859) where the word pôle does not even appear.

I have wondered if Neumann was perhaps inspired by prior uses of “pole” for homographies or inversions, $$f(z)=\frac{az+b}{cz+d}\qquad\text{or}\qquad f(z) = a+\frac k{\bar z-\bar a},$$ (presumably then traceable to Stubbs, Chasles, Servois,...) but couldn't find solid evidence.

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.

• According to the online etymological dictionary, the English word pole derives from the Latin word palus meaning stake. So the term could be handed down to us from when mathematicians still wrote in Latin. The term could also derive from Greek, polos (axis of a sphere), or it could be a coincidence. Simply because the direct cognate of a word in German doesn't correspond to the current meaning of an English word doesn't mean the words didn't have similar meanings long ago. The English word table is related to the German word Tafel even though Tafel only infrequently means table. – Peter Luthy Sep 22 '10 at 18:11
• A word derived from the Latin palus=stake does exist in German: Pfahl. In contrast, the Greek word polos becomes polus in Latin. So it appears that the North pole and the pole that is used to mark it actually have different etymological origins. – Michael Renardy Aug 13 '17 at 1:55

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.)

• This is a nice explanation, but I don't think that this is the correct one since in French (and German) the translation of your use of pole would not be "pôle" (or "Pol"). I guess it is rather related to the poles of a magnet. – Martin O Sep 21 '10 at 23:21
• Perhaps an explanation that would also make sense in French and German is that the value of a function at a pole is equal to infinity, the North pole of the Riemann sphere. – Faisal Sep 21 '10 at 23:52
• Upon reading the question, I figured something very close to Faisal's comment above. Poles are where $f(z)$ is the (North) Pole in the Riemann sphere. This, by the way, would make sense also in Italian (where "polo" is used both for Earth's poles and for, well, poles of meromorphic functions, whereas the "tall pole" in anon's answer is a "palo"). – Andrea Mori Sep 22 '10 at 8:12
• On second thought, I'm not so sure. For instance because the sphere has two poles (and for the Riemann sphere interpretation a zero should be called a "southern pole"). So I thought that maybe the two names are not directly related and they both derive from some other older meaning of the term. So, a relevant side question may be "why the Earth's poles are called such and who was the first to name them so?". Is a question for cartographers too off topic for MO? :-) – Andrea Mori Sep 22 '10 at 8:16
• @Martin: According to the online etymological dictionary, the English word pole derives from the Latin word palus meaning stake. So the term could be handed down to us from when mathematicians still wrote in Latin. The term could also derive from Greek, polos (axis of a sphere), or it could be a coincidence. Simply because the direct cognate of a word in German doesn't correspond to the current meaning of an English word doesn't mean the words didn't have similar meanings long ago. The English word table is related to the German word Tafel even though Tafel only infrequently means table. – Peter Luthy Sep 22 '10 at 18:09

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.

• To add to this, I have looked at the pages referred to for other words and they seem well-researched to me (who is not a mathematical historian). – Torsten Ekedahl Nov 7 '10 at 5:41
• "When a function u is holomorphic in a certain part of the plane, except for one point z_1, where it becomes infinite, similar to the function 1/u which remains holomorphic at the neighborhood of that point, we say that this point is a pole or an infinity of the function u." In the earlier paper, it's called "an infinity of finite degree". – Yuval Filmus Nov 7 '10 at 6:47
• I posted this as a comment to the main question a while back, but it apparently got buried... – j.c. Nov 7 '10 at 12:45
• @jc: I haven't noticed your previous comment, sorry. I'm making this answer Community Wiki. – Andrey Rekalo Nov 7 '10 at 12:56

This picture

should make it clear why they are called poles.

• These pictures used to be included in many complex analysis texts a few decades ago. I haven't encountered them in any of the recent book, though. – Andrés E. Caicedo Sep 22 '10 at 4:34