2006/05/24 by Bleher, Pavel, Mallison, Robert
#30D20 #30E10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.math-ph/0605066
We derive the large n asymptotics of zeros of sections of a generic exponential sum. We divide all the zeros of the n-th section of the exponential sum into ``genuine zeros'', which approach, as n→∞, the zeros of the exponential sum, and ``spurious zeros'', which go to infinity as n→∞. We show that the spurious zeros, after scaling down by the factor of n, approach a ``rosette'', a finite collection of curves on the complex plane, resembling the rosette. We derive also the large n asymptotics of the ``transitional zeros'', the intermediate zeros between genuine and spurious ones. Our results give an extension to the classical results of Szegö about the large n asymptotics of zeros of sections of the exponential, sine, and cosine functions.