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
$$
\Pi(a;r) = \sum_{\nu=0}^\infty c_\nu (z-a)^\nu 
$$
with centre $a$ and radius of convergence $r>0$ along all possible paths belonging to a given domain $D$ of the complex plane $\mathbf{C}$, $a \in D$. Thus, a branch of an analytic function is defined by the element $\Pi(a;r)$ and by the domain $D$. In calculations one usually employs only single-valued, or regular, branches of analytic functions, which need not exist for every domain $D$ belonging to the domain of existence of the [[complete analytic function]]. For instance, in the cut complex plane $D = \mathbf{C} \setminus \{ z = x : -\infty < x \le 0 \}$ the multi-valued analytic function $w = \mathrm{Ln}(z)$ has the regular branch
$$
w = \mathrm{Ln}(z) = \ln |z| + i \arg z\,,\ \ \ |\arg z| < \pi
$$
which is the principal value of the logarithm, whereas in the annulus $D = \{ z : 1 < |z| < 2 \}$ it is impossible to isolate a regular branch of the analytic function $w = \mathrm{Ln}(z)$.

====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>

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{{SourceattributionEOM|Branch of an analytic function}}
[[Category:Mathematics]]