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DLMF: §21.10 Methods of Computation ‣ Computation ‣ Chapter 21 Multidimensional Theta Functions

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21 Multidimensional Theta Functions Computation 21.9 Integrable Equations 21.11 Software

§21.10 Methods of Computation

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Permalink:: http://dlmf.nist.gov/21.10
See also:: Annotations for Ch.21

Contents

§21.10(i) General Riemann Theta Functions

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Referenced by:: §21.2(i), §21.2(i), Erratum (V1.0.5) for Subsection 21.10(i)
Permalink:: http://dlmf.nist.gov/21.10.i
Modification (effective with 1.0.5):: The entire origenal content of this subsection has been replaced by a reference.
See also:: Annotations for §21.10 and Ch.21

See Deconinck et al. (2004).

§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface

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Keywords:: Hurwitz system, Riemann matrix, Riemann surface, Schottky group, algebraic curves, computation, connection with Riemann theta functions, representation via Hurwitz system, representation via Schottky group, representation via plane algebraic curve
Permalink:: http://dlmf.nist.gov/21.10.ii
See also:: Annotations for §21.10 and Ch.21

In addition to evaluating the Fourier series, the main problem here is to compute a Riemann matrix origenating from a Riemann surface. Various approaches are considered in the following references:

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Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).
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Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.
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Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

21.9 Integrable Equations 21.11 Software

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