Quadratic Hermite-Pade approximation to the exponential function: a Riemann-Hilbert approach

Details Quadratic Hermite-Pade approximation to the exponential function: a Riemann-Hilbert approach Quadratic Hermite-Pade approximation to the exponential function: a Riemann-Hilbert approach

2003-02-28

arxiv.org/abs/math/0302357v1

Mathematics

Classical Analysis and ODEs

60 pages, 13 figures

Scientific paper

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements recent results of Herbert Stahl.

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