via Weierstrass elliptic functions

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1: 23.20 Mathematical Applications

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§23.20(iii) Factorization

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2: 20.11 Generalizations and Analogs

… ►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). …

3: 23.3 Differential Equations

… ►The lattice invariants are defined by … ►The lattice roots satisfy the cubic equation …The discriminant (§1.11(ii)) is given by … ►However, given any pair of generators $2{\omega }_1$, $2{\omega }_3$ of $𝕃$, and with ${\omega }_2$ defined by (23.2.1), we can identify the $e_j$ individually, via … ►

§23.3(ii) Differential Equations and Derivatives

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4: 23.22 Methods of Computation

§23.22 Methods of Computation

… ►For $\mathrm{\wp }\left(z\right)$ we apply (23.6.2) and (23.6.5), generating all needed values of the theta functions by the methods described in §20.14. ►The functions $\zeta \left(z\right)$ and $\sigma \left(z\right)$ are computed in a similar manner: the former by replacing $u$ and $z$ in (23.6.13) by $z$ and $\pi z/(2{\omega }_1)$, respectively, and also referring to (23.6.8); the latter by applying (23.6.9). … ►

§23.22(ii) Lattice Calculations

… ►Suppose that the invariants $g_2=c$, $g_3=d$, are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve (§23.20(ii)). …

5: Software Index

… ►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …