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Logarithmic branch point

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branch point of infinite order

A special form of a branch point of an analytic function of one complex variable , when for no finite number of successive circuits in the same direction about the analytic continuation of some element of returns to the original element. More precisely, an isolated singular point is called a logarithmic branch point for if there exist: 1) an annulus in which can be analytically continued along any path; and 2) a point and an element of in the form of a power series with centre and radius of convergence , the analytic continuation of which along the circle , taken arbitrarily many times in the same direction, never returns to the original element . In the case of a logarithmic branch point at infinity, , instead of one must consider a neighbourhood . Logarithmic branch points belong to the class of transcendental branch points (cf. Transcendental branch point). The behaviour of the Riemann surface of a function in the presence of a logarithmic branch point is characterized by the fact that infinitely many sheets of the same branch of are joined over ; this branch is defined in or by the elements .

See also Singular point of an analytic function.

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)


Comments

The function has a logarithmic branch point at , where is the (multiple-valued) logarithmic function of a complex variable.

References

[a1] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8
How to Cite This Entry:
Logarithmic branch point. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Logarithmic_branch_point&oldid=11411
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098