Branch of an analytic function

The result of 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

[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)