Algebraic branch point

algebraic singular point

An isolated branch point $a$ of finite order of an analytic function $f(z)$, having the property that the limit $\lim_{z\to a}f(z)$ exists for any regular element of continuation of $f$ in a domain for which $a$ is a boundary point. More exactly, a singular point $a$ in the complex $z$-plane for the complete analytic function $f(z)$, under continuation of some regular element $e_0$ of this function with centre $z_0$ along paths passing through $a$, is called an algebraic branch point if it fulfills the following conditions: 1) There exists a positive number $\rho$ such that the element $e_0$ may be extended along an arbitrary continuous curve lying in the annulus $D=\{z:0<|z-a|<\rho\}$; 2) there exists a positive integer $k>1$ such that if $z_1$ is an arbitrary point of $D$, the analytic continuation of the element $e_0$ in $D$ yields exactly $k$ different elements of the function $f(z)$ with centre $z_1$; if $e_1$ is an arbitrary element with centre $z_1$, all the remaining $k-1$ elements with centre $z_1$ can be obtained by analytic continuation along closed paths around the point $a$; and 3) the values at the points $z$ of $D$ of all elements which are obtainable from $e_0$ by continuation in $D$ tend to a definite, finite or infinite, limit as $z$ tends to $a$ while remaining in D.

The number $k-1$ is said to be the order of the algebraic branch point. All branches of the function $f(z)$ obtainable by analytic continuation of the element $e_0$ in the annulus $D$ may be represented in a deleted neighbourhood of $a$ by a generalized Laurent series (Puiseux series):

$$f(z)=\sum_{n=-m}^\infty c_n(z-a)^{n/k},\quad m\geq0.$$

The point at infinity, $a=\infty$, is called an algebraic branch point for a function $f(z)$ if the point $b=0$ is an algebraic branch point of the function $g(w)=f(1/w)$.

There may exist several (and even an infinite number of) different algebraic branch points and regular points of a complete analytic function with a given affix $a$.

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt.8 (Translated from Russian)
[2]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 4 , Springer (1968)