Algebraic branch point
algebraic singular point
An isolated branch point of finite order of an analytic function
, having the property that the limit
exists for any regular element of continuation of
in a domain for which
is a boundary point. More exactly, a singular point
in the complex
-plane for the complete analytic function
, under continuation of some regular element
of this function with centre
along paths passing through
, is called an algebraic branch point if it fulfills the following conditions: 1) There exists a positive number
such that the element
may be extended along an arbitrary continuous curve lying in the annulus
; 2) there exists a positive integer
such that if
is an arbitrary point of
, the analytic continuation of the element
in
yields exactly
different elements of the function
with centre
; if
is an arbitrary element with centre
, all the remaining
elements with centre
can be obtained by analytic continuation along closed paths around the point
; and 3) the values at the points
of
of all elements which are obtainable from
by continuation in
tend to a definite, finite or infinite, limit as
tends to
while remaining in D.
The number is said to be the order of the algebraic branch point. All branches of the function
obtainable by analytic continuation of the element
in the annulus
may be represented in a deleted neighbourhood of
by a generalized Laurent series (Puiseux series):
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The point at infinity, , is called an algebraic branch point for a function
if the point
is an algebraic branch point of the function
.
There may exist several (and even an infinite number of) different algebraic branch points and regular points of a complete analytic function with a given affix .
References
[1] | A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt.8 (Translated from Russian) |
[2] | A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 4 , Springer (1968) |
Algebraic branch point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_branch_point&oldid=43558