# Branch point

singular point of multi-valued character

An isolated singular point $a$ of an analytic function $f(z)$ of one complex variable $z$ such that the analytic continuation of an arbitrary function element of $f(z)$ along a closed path which encircles $a$ yields new elements of $f(z)$. More exactly, $a$ is said to be a branch point if there exist: 1) an annulus $V= \{ {z } : {0 < | z - a | < \rho } \}$ in which $f(z)$ can be analytically extended along any path; 2) a point $z _ {1} \in V$ and some function element of $f(z)$ represented by a power series

$$\Pi (z _ {1} ; r) = \ \sum _ {v = 0 } ^ \infty c _ {v} (z - z _ {1} ) ^ {v}$$

with centre $z _ {1}$ and radius of convergence $r > 0$, the analytic continuation of which along the circle $| z - a | = | z _ {1} - a |$, going around the path once in, say, the positive direction, yields a new element $\Pi ^ { \prime } (z _ {1} ; r ^ \prime )$ different from $\Pi (z _ {1} ; r)$. If, after a minimum number $k > 1$ of such rounds the initial element $\Pi (z _ {1} ; r)$ is again obtained, this is also true of all elements of the branch (cf. Branch of an analytic function) of $f(z)$ defined in $V$ by the element $\Pi (z _ {1} ; r)$. In such a case $a$ is a branch point of finite order $k - 1$ of this branch. In a punctured neighbourhood $V$ of a branch point $a$ of finite order this branch is represented by a generalized Laurent series, or Puiseux series:

$$\tag{1 } f (z) = \ \sum _ {v = - \infty } ^ { {+ } \infty } b _ {v} (z - a) ^ {v/k} , \ z \in V.$$

If $a = \infty$ is an improper branch point of a finite order, then the branch of $f(z)$ is representable in some neighbourhood $V ^ { \prime } = \{ {z } : {| z | > \rho } \}$ by an analogue of the series (1):

$$\tag{2 } f (z) = \ \sum _ {v = - \infty } ^ { {+ } \infty } b _ {v} z ^ {-v/k} , \ z \in V ^ \prime .$$

The behaviour of the Riemann surface $R$ of $f(z)$ over a branch point of finite order $a$ is characterized by the fact that $k$ sheets of the branch of $f(z)$ defined by the element $\Pi (z _ {1} ; r)$ come together over $a$. At the same time the behaviour of other branches of $R$ over $a$ may be altogether different.

If the series (1) or (2) contains only a finite number of non-zero coefficients $b _ {v}$ with negative indices $v$, $a$ is an algebraic branch point or an algebraic singular point. Such a branch point of finite order is also characterized by the fact that as $z \rightarrow a$ in whatever manner, the values of all elements of the branch defined by $\Pi (z _ {1} ; r)$ in $V$ or $V ^ { \prime }$ tend to a definite finite or infinite limit.

Example: $f (z) = z ^ {1/k}$, where $k > 1$ is a natural number, $a = 0, \infty$.

If the series (1) or (2) contain an infinite number of non-zero coefficients $b _ {v}$ with negative indices $v$, the branch points of finite order $a$ belong the class of transcendental branch points.

Example: $f (z) = \mathop{\rm exp} (1/z) ^ {1/k}$, where $k > 1$ is a natural number, $a = 0$.

Finally, if it is impossible to return to the initial element after a finite number of turns, $a$ is said to be a logarithmic branch point or a branch point of infinite order, and is also a transcendental branch point.

Example: $f(z) = \mathop{\rm Ln} z, a = 0, \infty$.

Infinitely many sheets of the branch of $f(z)$ defined by the element $\Pi (z _ {1} ; r)$ come together over a logarithmic branch point.

In the case of an analytic function of several complex variables $f(z)$, $z = (z _ {1} \dots z _ {n} )$, $n \geq 2$, a point $a$ of the space $\mathbf C ^ {n}$ or $\mathbf C P ^ {n}$ is said to be a branch point of order $m$, $1 \leq m \leq \infty$, if it is a branch point of order $m$ of the, generally many-sheeted, domain of holomorphy of $f(z)$. Unlike in the case $n=1$, branch points, just like other singular points of analytic functions (cf. Singular point), cannot be isolated if $n \geq 2$.

How to Cite This Entry:
Branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branch_point&oldid=46148
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article