general elliptic functions

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21: 19.16 Definitions

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§19.16(i) Symmetric Integrals

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§19.16(ii) $R_{-a}(𝐛;𝐳)$

… ►The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … ►For generalizations and further information, especially representation of the $R$-function as a Dirichlet average, see Carlson (1977b). … ►

§19.16(iii) Various Cases of $R_{-a}(𝐛;𝐳)$

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22: 15.17 Mathematical Applications

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§15.17(ii) Conformal Mappings

… ►Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. … ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …

23: 23.7 Quarter Periods

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23.7.1 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_1\right)=e_1+\sqrt{(e_1-e_3)(e_1-e_2)}=e_1+{\omega }_1^{-2}{(K\left(k\right))}^2{k}^{\prime },$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E1

Encodings:

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See also:

Annotations for §23.7 and Ch.23

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23.7.2 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_2\right)=e_2-\mathrm{i}\sqrt{(e_1-e_2)(e_2-e_3)}=e_2-\mathrm{i}{\omega }_1^{-2}{(K\left(k\right))}^{2}k{k}^{\prime },$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

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23.7.3 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_3\right)=e_3-\sqrt{(e_1-e_3)(e_2-e_3)}=e_3-{\omega }_1^{-2}{(K\left(k\right))}^{2}k,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

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24: Bibliography B

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Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.

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Notes:

Algol 60 programs for the incomplete elliptic integral of the third kind, and for a general complete elliptic integral, are included.

External Links:

ISSN 0029-599X, Document, MathReview, ZentralBlatt

Referenced by:

§19.2(iii), §19.36(ii), §19.36(ii), §19.39(ii), §19.39(iii), R. Bulirsch (1965a).

Encodings:

BibTeX

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25: 23.22 Methods of Computation

§23.22 Methods of Computation

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§23.22(i) Function Values

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§23.22(ii) Lattice Calculations

… ►Then a pair of generators $2{\omega }_1$ and $2{\omega }_3$ can be chosen in an almost canonical way as follows. … ►

If $d=0$, then

23.22.2 $$2{\omega }_1=-2\mathrm{i}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2}{2\sqrt{\pi }{c}^{1/4}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§22.20(iv), §23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 4 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when $c>0$ and ${\omega }_1>0$.

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26: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

… ►More generally in (19.14.4), … ►

§19.14(ii) General Case

►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. …

27: Software Index

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	Open Source													With Book						Commercial
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22 Jacobian Elliptic Functions
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►‘✓’ 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. … ►

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. …

28: Bibliography W

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Evaluating elliptic functions and their inverses. Comput. Math. Appl. 39 (3-4), pp. 131–136.

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External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§22.20(ii), §22.20(v).

Encodings:

BibTeX

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Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§4.12.

Encodings:

BibTeX

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Elliptic Functions According to Eisenstein and Kronecker. Classics in Mathematics, Springer-Verlag, Berlin.

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Notes:

Reprint of the 1976 original

External Links:

ISBN 3-540-65036-9, MathReview, ZentralBlatt

Referenced by:

§22.12, §23.8(ii).

Encodings:

BibTeX

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On a generalization of the functions $\mathrm{ber}$ x, $\mathrm{bei}$ x, $\ker$ x, $\mathrm{kei}$ x. Quart. J. Pure Appl. Math. 42, pp. 316–342.

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External Links:

ZentralBlatt

Referenced by:

§10.61(iii), §10.61(iv), §10.65(i), §10.65(ii), §10.65(iii), §10.68(iii).

Encodings:

BibTeX

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The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.46.

Encodings:

BibTeX

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29: 22 Jacobian Elliptic Functions

Chapter 22 Jacobian Elliptic Functions

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30: 23.18 Modular Transformations

§23.18 Modular Transformations

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Elliptic Modular Function

… ►Here e and o are generic symbols for even and odd integers, respectively. … ►

Dedekind’s Eta Function

… ►where the square root has its principal value and …