RT Journal Article
SR Electronic
T1 Universal oscillations of high derivatives
JF Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
JO PROC R SOC A
FD The Royal Society
SP 1735
OP 1751
DO 10.1098/rspa.2005.1446
VO 461
IS 2058
A1 Berry, M.V
YR 2005
UL http://rspa.royalsocietypublishing.org/content/461/2058/1735.abstract
AB 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 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.