The Riemann surfaces of a function and its fractional integral

Abstract

2. Transformation of Riemann surfaces. Theorem 1. Let f(z) be analytic within a circle with centre at a, and which contains I in its interior. Then a is a branch-point of D~ (la)f(z) for non-integral values of X. If A is a rational fraction rjg expressed in its lowest terms, then a is the vertex of a cycle of s roots. If A is irrational or complex, then a is the vertex of an infinite number of roots.