View source for Branch of an analytic function

The result of [[Analytic continuation|analytic continuation]] of a given element of an analytic function represented by a power series

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with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b0174902.png" /> and radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b0174903.png" /> along all possible paths belonging to a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b0174904.png" /> of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b0174905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b0174906.png" />. Thus, a branch of an analytic function is defined by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b0174907.png" /> and by the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b0174908.png" />. In calculations one usually employs only single-valued, or regular, branches of analytic functions, which exist not for all domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b0174909.png" /> belonging to the domain of existence of the [[Complete analytic function|complete analytic function]]. For instance, in the cut complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b01749010.png" /> the multi-valued analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b01749011.png" /> has the regular branch

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which is the principal value of the logarithm, whereas in the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b01749013.png" /> it is impossible to isolate a regular branch of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017490/b01749014.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Hurwitz,   R. Courant,   "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1 Chapt. 3; 2, Chapt. 4''' , Springer  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>