A Darboux-Type Theorem for Slowly Varying Functions

✍ Scribed by B.L.J. Braaksma; D. Stark

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 556 KB
• Volume: 77
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1996.2714

✦ Synopsis

For functions g(z) satisfying a slowly varying condition in the complex plane, we find asymptotics for the Taylor coefficients of the function

when :>0. As applications we find asymptotics for the number of permutations with cycle lengths all lying in a given set S, and for the number having unique cycle lengths.

1997 Academic Press

1. Introduction

Often the asymptotics of a function's Taylor coefficients can be determined by the behavior of the function near its singularities of smallest modulus. Information of these asymptotics is useful in probability, combinatorics and theoretical computer science. We briefly state three theorems that obtain such asymptotic information before presenting our results. More detailed information on all three is given in [6].