Difference between revisions of "Transcendental branch point"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
− | + | <!-- | |
+ | t0936001.png | ||
+ | $#A+1 = 11 n = 0 | ||
+ | $#C+1 = 11 : ~/encyclopedia/old_files/data/T093/T.0903600 Transcendental branch point | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
− | + | {{TEX|auto}} | |
+ | {{TEX|done}} | ||
− | + | ''of an analytic function $ f ( z) $'' | |
− | + | A [[Branch point|branch point]] that is not an [[Algebraic branch point|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|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 $. | ||
− | with an infinite number of non-zero coefficients | + | In the first case the function $ f ( z) $ |
+ | can be expanded in a neighbourhood of $ a $ | ||
+ | in the form of a [[Puiseux series|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==== | ====References==== | ||
<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> | <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> |
Revision as of 08:26, 6 June 2020
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=49007