Algebraic branch point
algebraic singular point
An isolated branch point 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) |
Algebraic branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_branch_point&oldid=43559