Regular Article
A Darboux-Type Theorem for Slowly Varying Functions

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Abstract

For functionsg(z) satisfying a slowly varying condition in the complex plane, we find asymptotics for the Taylor coefficients of the function[formula]whenα>0. As applications we find asymptotics for the number of permutations with cycle lengths all lying in a given setS, and for the number having unique cycle lengths.

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Part of this author's work was done at the University of Southern California, and he thanks the USC Department of Mathematics for their hospitality and in particular Professor W. A. Harris, Jr.

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The author thanks the University of Groningen Mathematics Department for a pleasant visit during which part of the work was completed.

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