Algebraic Branch Point

A branch point whose neighborhood of values wrap around the range a finite number of times  as their complex arguments  varies from 0 to a multiple of  is called an algebraic branch point of order . Such points correspond to the point  under functions of the form .

Formally, an algebraic branch point is a singular boundary point of one sheet of a multivalued function about which a finite number  of distinct sheets hang together like the surface for  at the origin and for which the domain of values affixed to these sheets in a neighborhood of , which can be developed in a series of the form

is such that only a finite number (or zero) negative power of  appear in the expansion (Knopp 1996, Part II, p. 143).

REFERENCES:

Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, Part II, pp. 142-143, 1996.