Universal oscillations of high derivatives

Differentiation generates oscillations. For the nth 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 saddlepoints. 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 ffiffiffi n p 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.