Logarithmic branch point

branch point of infinite order

A special form of a branch point $a$ of an analytic function $f ( z)$ of one complex variable $z$, when for no finite number of successive circuits in the same direction about $a$ the analytic continuation of some element of $f ( z)$ returns to the original element. More precisely, an isolated singular point $a$ is called a logarithmic branch point for $f ( z)$ if there exist: 1) an annulus $V = \{ {z } : {0 < | z - a | < \rho } \}$ in which $f ( z)$ can be analytically continued along any path; and 2) a point $z _ {1} \in V$ and an element of $f ( z)$ in the form of a power series $\Pi ( z _ {1} ; r ) = \sum _ {\nu = 0 } ^ \infty c _ \nu ( z - z _ {1} ) ^ \nu$ with centre $z _ {1}$ and radius of convergence $r > 0$, the analytic continuation of which along the circle $| z - a | = | z _ {1} - a |$, taken arbitrarily many times in the same direction, never returns to the original element $\Pi ( z _ {1} ; r )$. In the case of a logarithmic branch point at infinity, $a = \infty$, instead of $V$ one must consider a neighbourhood $V ^ \prime = \{ {z } : {| z | > \rho } \}$. Logarithmic branch points belong to the class of transcendental branch points (cf. Transcendental branch point). The behaviour of the Riemann surface $R$ of a function $f ( z)$ in the presence of a logarithmic branch point $a$ is characterized by the fact that infinitely many sheets of the same branch of $R$ are joined over $a$; this branch is defined in $V$ or $V ^ \prime$ by the elements $\Pi ( z _ {1} ; r )$.

See also Singular point of an analytic function.

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)

Comments

The function $\mathop{\rm Ln} ( z - z _ {0} )$ has a logarithmic branch point at $z _ {0}$, where $\mathop{\rm Ln}$ is the (multiple-valued) logarithmic function of a complex variable.

References

[a1]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8