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Computing Riemann theta functions

Authors: Bernard Deconinck, Matthias Heil, Alexander Bobenko, Mark van Hoeij and Marcus Schmies
Translated by:
Journal: Math. Comp. 73 (2004), 1417-1442
MSC (2000): Primary 14K25, 30E10, 33F05, 65D20
Published electronically: December 19, 2003
MathSciNet review: 2047094
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Abstract: The Riemann theta function is a complex-valued function of $g$ complex variables. It appears in the construction of many (quasi-)periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula is used to construct a uniform approximation formula, again with arbitrary precision.

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Additional Information

Bernard Deconinck
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523-1874

Matthias Heil
Affiliation: Fachbereich Mathematik, Technische Universität Berlin, Strasse des 17.Juni 136, 10623 Berlin, Germany

Alexander Bobenko
Affiliation: Fachbereich Mathematik, Technische Universität Berlin, Strass des 17.Juni 136, 10623 Berlin, Germany

Mark van Hoeij
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306

Marcus Schmies
Affiliation: Fachbereich Mathematik, Technische Universität Berlin, Strass des 17.Juni 136, 10623 Berlin, Germany

Keywords: Riemann theta function, pointwise approximation, uniform approximation
Received by editor(s): June 7, 2002
Published electronically: December 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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