ZEROS OF SECTIONS OF EXPONENTIAL SUMS

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 Szego about the large n asymptotics of zeros of sections of the exponential, sine, and cosine functions.

[1]  R. Varga,et al.  Zeros of Sections of Power Series , 1983 .

[2]  Tanja Bergkvist,et al.  On polynomial eigenfunctions for a class of differential operators , 2002 .

[3]  Kenneth R. Cramer,et al.  Complex Zeros of the Error Function and of the Complementary Error Function , 1973 .

[4]  Kenneth B. Huber Department of Mathematics , 1894 .

[5]  Richard S. Varga,et al.  Zeros of the partial sums of $\cos(z)$ and $\sin(z)$ II , 2001, Numerische Mathematik.

[6]  Exponential Gelfond–Khovanskii formula in dimension one , 2003, math/0312433.

[7]  R. Langer The asymptotic location of the roots of a certain transcendental equation , 1929 .

[8]  Richard S. Varga,et al.  Asymptotics for the zeros of the partial sums of ez. II , 1991 .

[9]  J. D. Buckholtz,et al.  A Characterization of the Exponential Series , 1966 .

[10]  K. Iverson The Zeros of the Partial Sums of e z , 1953 .

[11]  Donald J. Newman,et al.  The zeros of the partial sums of the exponential function , 1972 .

[12]  A. Kuijlaars,et al.  Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter , 2002, math/0205175.

[13]  Stephen M. Zemyan,et al.  On the Zeroes of the Nth Partial Sum of the Exponential Series , 2005, Am. Math. Mon..

[14]  M. Kappert On the zeros of the partial sums of $\cos(z)$ and $\sin(z)$ , 1996 .

[15]  Brian Conrey,et al.  On the zeros of the Taylor polynomials associated with the exponential function , 1988 .

[16]  Richard S. Varga,et al.  Zero-Free Parabolic Regions for Sequences of Polynomials , 1976 .

[17]  B. Levin,et al.  Distribution of zeros of entire functions , 1964 .

[18]  William M. Y. Goh,et al.  On the zero attractor of the Euler polynomials , 2007, Adv. Appl. Math..

[19]  Richard S. Varga,et al.  Zeros of the partial sums of cos (z) and sin (z). I , 2000, Numerical Algorithms.

[20]  R. Varga,et al.  The Szego curve, zero distribution and weighted approximation , 1997 .