Universal oscillations of high derivatives

Abstract

Differentiation generates oscillations. For the n th derivative f ( n ,  t ) of a function f ( t ) that is analytic in a strip, including the real t -axis, the oscillations occupy a t interval that gets larger as n increases. The oscillations are studied in detail using integral representations and large- n asymptotics. For functions with singularities (poles or branch-points) in the complex t -plane, the oscillations of high derivatives are determined by the singularities; for entire functions, the oscillations originate in complex saddle-points. In a wide class of cases, the oscillations are contained in a Gaussian envelope in the t interval where f ( n ,  t ) is largest, with the envelope including about oscillations. Examples of the universal oscillations are given for f ( t ) with a simple pole, competing branch-points, a single saddle, competing pole and saddle and where the zeros are confined to half the real axis.