Algebraic Branch Point

A branch point whose neighborhood of values wrap around the range a finite number of times p as their complex arguments theta varies from 0 to a multiple of 2pi is called an algebraic branch point of order p. Such points correspond to the point z=0 under functions of the form f(z)=z^(q/p).

Formally, an algebraic branch point is a singular boundary point of one sheet of a multivalued function about which a finite number p of distinct sheets hang together like the surface for z^(1/p) at the origin and for which the domain of values affixed to these p sheets in a neighborhood of z_0, which can be developed in a series of the form


is such that only a finite number (or zero) negative power of (z-z_0)^(1/p) appear in the expansion (Knopp 1996, Part II, p. 143).

See also

Branch Point, Logarithmic Branch Point

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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.

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Algebraic Branch Point

Cite this as:

Weisstein, Eric W. "Algebraic Branch Point." From MathWorld--A Wolfram Web Resource.

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