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A Fractional Partial Differential Equation for Theta Functions

In: Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

Author

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  • Rafael G. Campos

    (Universidad de Quintana Roo, División de Ciencias e Ingeniería, Departamento de Ciencias)

Abstract

We find that theta functions are solutions of a fractional partial differential equation that generalizes the diffusion equation. This equation is the limit of a sequence of differential equations for the partial sums of theta functions where the fractional derivatives are given as differentiation matrices for trigonometric polynomials in their Fourier representation, i.e., given as similarities of diagonal matrices under the ordinary discrete Fourier transform. This fact enables the fast numerical computation of fractional partial derivatives of theta functions and elliptic integrals.

Suggested Citation

  • Rafael G. Campos, 2018. "A Fractional Partial Differential Equation for Theta Functions," Springer Books, in: Gert-Martin Greuel & Luis Narváez Macarro & Sebastià Xambó-Descamps (ed.), Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, pages 579-591, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-96827-8_26
    DOI: 10.1007/978-3-319-96827-8_26
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