# Transcendental branch point

of an analytic function $f ( z)$

A branch point that is not an algebraic branch point. In other words, it is either a branch point $a$ of finite order $k > 0$ at which, however, there does not exist a finite or infinite limit

$$\lim\limits _ {\begin{array}{c} z \rightarrow a \\ z \neq a \end{array} } f ( z),$$

or a logarithmic branch point of infinite order. For example, the first possibility is realized at the transcendental branch point $a = 0$ for the function $\mathop{\rm exp} ( 1/z ^ {1/k} )$, the second for the function $\mathop{\rm ln} z$.

In the first case the function $f ( z)$ can be expanded in a neighbourhood of $a$ in the form of a Puiseux series

$$f ( z) = \sum _ {n = - \infty } ^ { {+ } \infty } c _ {n} ( z - a) ^ {n/k}$$

with an infinite number of non-zero coefficients $c _ {n}$ with negative indices.

#### References

 [1] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Transcendental branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transcendental_branch_point&oldid=49007
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article