Difference between revisions of "Transcendental branch point"
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− | f ( z) = \ | + | f ( z) = \sum_{n = - \infty } ^ {+\infty } c _ {n} ( z - a) ^ {n/k} |
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, | + | <table> |
+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) (Translated from Russian)</TD></TR> | ||
+ | </table> |
Latest revision as of 08:30, 20 January 2024
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) |
Transcendental branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transcendental_branch_point&oldid=55231