Welcome to MathOverflow

Question

18

2

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?

edited Sep 21 2010 at 21:17

Yemon Choi
8,456●1●19●45

asked Sep 21 2010 at 19:55

Trevor C
181●3

8

It's because they correspond to the north pole when considered as holomorphic functions onto the riemann sphere. – muad Sep 21 2010 at 20:50

7

Insert your favorite Polish joke here. – Nate Eldredge Sep 21 2010 at 21:21

14

I, too, would like this question stay open. Please, count it as a vote against closing per old meta discussion. – Victor Protsak Sep 21 2010 at 22:21

12

@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 2010 at 23:20

21

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 2010 at 0:10

show 14 more comments

anon · Answer 1 · 2010-09-21 23:00:00Z UTC

9

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.)

answered Sep 21 2010 at 23:00

anon
189●4

9

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 2010 at 23:21

3

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 2010 at 23:52

1

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 2010 at 8:12

1

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 2010 at 8:16

2

@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 2010 at 18:09

show 3 more comments

Greg Graviton · Answer 2 · 2010-09-22 09:36:22Z UTC

7

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.

answered Sep 22 2010 at 9:36

Greg Graviton
758●1●4●10

1

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 2010 at 18:11

Richard Borcherds · Answer 3 · 2010-09-22 04:19:23Z UTC

5

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

edited Sep 22 2010 at 4:19

answered Sep 22 2010 at 2:32

Richard Borcherds
14.1k●2●54●82

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. – Andres Caicedo Sep 22 2010 at 4:34

Andrey Rekalo · Answer 4 · 2010-11-07 04:48:06Z UTC

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.

What is the naming reason of poles in complex analysis?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

4 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.