Branch of an analytic function

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The result of analytic continuation of a given element of an analytic function represented by a power series

with centre and radius of convergence along all possible paths belonging to a given domain of the complex plane , . Thus, a branch of an analytic function is defined by the element and by the domain . In calculations one usually employs only single-valued, or regular, branches of analytic functions, which exist not for all domains belonging to the domain of existence of the complete analytic function. For instance, in the cut complex plane the multi-valued analytic function has the regular branch

which is the principal value of the logarithm, whereas in the annulus it is impossible to isolate a regular branch of the analytic function .


[1] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 Chapt. 3; 2, Chapt. 4 , Springer (1964)
[2] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Branch of an analytic function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article