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Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory pp 269–293Cite as

Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms

Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

We describe algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for arbitrary complex variables. Implementations in ball arithmetic are available in the open source Arb library. We discuss the algorithms from a concrete implementation point of view, with focus on performance at tens to thousands of digits of precision.

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Fig. 1
Fig. 2

Notes

  1. 1.

    Available at http://arblib.org. The functionality for modular forms and elliptic functions can be found in the acb_modular (http://arblib.org/acb_modular.html) and acb_elliptic (http://arblib.org/acb_elliptic.html) modules.

  2. 2.

    Of course, for applications that do not require rigorous error bounds, all the algorithms can just as well be implemented in ordinary floating-point arithmetic.

  3. 3.

    https://dlmf.nist.gov/.

  4. 4.

    The number is somewhat smaller if the series is truncated optimally using a relative rather than an absolute tolerance.

  5. 5.

    We give the inverse form of the transformation.

  6. 6.

    For \(\varPi \), Mathematica restricts this quasiperiodicity relation to hold only for \(-1 \le n \le 1\).

  7. 7.

    This is an algebraic simplification, so we can take \(E_1 = 0\) even if the input argument are represented by inexact balls.

  8. 8.

    At precision up to about 1000 digits, the elementary functions in Arb are significantly faster than the AGM due to using precomputed lookup tables and many low-level optimizations [14].

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Acknowledgements

The author thanks the organizers of the KMPB Conference on Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory for the invitation to present this work at DESY in October 2017 and for the opportunity to publish this extended review in the post-conference proceedings.

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Authors and Affiliations

  1. INRIA – LFANT, CNRS – IMB – UMR 5251, Université de Bordeaux, 33400, Talence, France

    Fredrik Johansson

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  1. Fredrik Johansson
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Correspondence to Fredrik Johansson .

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Editors and Affiliations

  1. Deutsches Elektronen-Synchrotron, DESY, Zeuthen, Germany

    Prof. Dr. Johannes Blümlein

  2. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Prof. Dr. Carsten Schneider

  3. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Prof. Dr. Peter Paule

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Johansson, F. (2019). Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_12

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  • DOI: https://doi.org/10.1007/978-3-030-04480-0_12

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