Logarithmic branch point

branch point of infinite order

A special form of a branch point $ a $ of an analytic function $ f ( z) $ of one complex variable $ z $, when for no finite number of successive circuits in the same direction about $ a $ the analytic continuation of some element of $ f ( z) $ returns to the original element. More precisely, an isolated singular point $ a $ is called a logarithmic branch point for $ f ( z) $ if there exist: 1) an annulus $ V = \{ {z } : {0 < | z - a | < \rho } \} $ in which $ f ( z) $ can be analytically continued along any path; and 2) a point $ z _ {1} \in V $ and an element of $ f ( z) $ in the form of a power series $ \Pi ( z _ {1} ; r ) = \sum _ {\nu = 0 } ^ \infty c _ \nu ( z - z _ {1} ) ^ \nu $ with centre $ z _ {1} $ and radius of convergence $ r > 0 $, the analytic continuation of which along the circle $ | z - a | = | z _ {1} - a | $, taken arbitrarily many times in the same direction, never returns to the original element $ \Pi ( z _ {1} ; r ) $. In the case of a logarithmic branch point at infinity, $ a = \infty $, instead of $ V $ one must consider a neighbourhood $ V ^ \prime = \{ {z } : {| z | > \rho } \} $. Logarithmic branch points belong to the class of transcendental branch points (cf. Transcendental branch point). The behaviour of the Riemann surface $ R $ of a function $ f ( z) $ in the presence of a logarithmic branch point $ a $ is characterized by the fact that infinitely many sheets of the same branch of $ R $ are joined over $ a $; this branch is defined in $ V $ or $ V ^ \prime $ by the elements $ \Pi ( z _ {1} ; r ) $.

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 $ \mathop{\rm Ln} ( z - z _ {0} ) $ has a logarithmic branch point at $ z _ {0} $, where $ \mathop{\rm Ln} $ is the (multiple-valued) logarithmic function of a complex variable.

References

[a1]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8