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):

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)