TITLE: What is the naming reason of poles in complex analysis?
QUESTION [26 upvotes]: 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?

REPLY [20 votes]: 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.