Branch of an analytic function
The result of analytic continuation of a given element of an analytic function represented by a power series
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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
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which is the principal value of the logarithm, whereas in the annulus it is impossible to isolate a regular branch of the analytic function
.
References
[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) |
Branch of an analytic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branch_of_an_analytic_function&oldid=16288