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Transcendental branch point

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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