general elliptic functions

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41: 23.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►►

$𝕃$	lattice in $ℂ$.
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$K\left(k\right)$, $K^{\prime }\left(k\right)$	complete elliptic integrals (§19.2(i)).
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$={\mathrm{e}}^{\mathrm{i}\pi \tau }$	nome.
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►The main functions treated in this chapter are the Weierstrass $\mathrm{\wp }$-function $\mathrm{\wp }\left(z\right)=\mathrm{\wp }\left(z|𝕃\right)=\mathrm{\wp }(z;g_2,g_3)$; the Weierstrass zeta function $\zeta \left(z\right)=\zeta \left(z|𝕃\right)=\zeta (z;g_2,g_3)$; the Weierstrass sigma function $\sigma \left(z\right)=\sigma \left(z|𝕃\right)=\sigma (z;g_2,g_3)$; the elliptic modular function $\lambda \left(\tau \right)$; Klein’s complete invariant $J\left(\tau \right)$; Dedekind’s eta function $\eta \left(\tau \right)$. …

42: 23.4 Graphics

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§23.4(i) Real Variables

►Line graphs of the Weierstrass functions $\mathrm{\wp }\left(x\right)$, $\zeta \left(x\right)$, and $\sigma \left(x\right)$, illustrating the lemniscatic and equianharmonic cases. … ►

§23.4(ii) Complex Variables

►Surfaces for the Weierstrass functions $\mathrm{\wp }\left(z\right)$, $\zeta \left(z\right)$, and $\sigma \left(z\right)$. Height corresponds to the absolute value of the function and color to the phase. …

43: 19 Elliptic Integrals

Chapter 19 Elliptic Integrals

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44: 23.8 Trigonometric Series and Products

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§23.8(i) Fourier Series

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§23.8(ii) Series of Cosecants and Cotangents

… ►where in (23.8.4) the terms in $n$ and $-n$ are to be bracketed together (the Eisenstein convention or principal value: see Weil (1999, p. 6) or Walker (1996, p. 3)). … ►

23.8.5 $${\eta }_1=\frac{{\pi }^2}{2{\omega }_1}\left(\frac{1}{6}+\sum _{n=1}^{\mathrm{\infty }}{\csc }^2\left(\frac{n\pi {\omega }_3}{{\omega }_1}\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.8(ii), §23.8 and Ch.23

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§23.8(iii) Infinite Products

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45: 23.5 Special Lattices

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§23.5(ii) Rectangular Lattice

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§23.5(iii) Lemniscatic Lattice

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§23.5(iv) Rhombic Lattice

… ►As a function of $\mathrm{\Im }{e}_3$ the root $e_1$ is increasing. … ►

§23.5(v) Equianharmonic Lattice

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46: Bibliography M

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An improved calculation of the general elliptic integral of the second kind in the neighbourhood of $x=0$ . Numer. Math. 25 (1), pp. 99–101.

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External Links:

ISSN 0029-599X, Document, MathReview (Roland Bulirsch), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

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The generalized integro-exponential function. Math. Comp. 44 (170), pp. 443–458.

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External Links:

ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

§8.19(v), §8.19(xi).

Encodings:

BibTeX

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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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External Links:

ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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Jacobian Elliptic Function Tables. Dover Publications Inc., New York.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§22.20(vii).

Encodings:

BibTeX

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47: 19.10 Relations to Other Functions

§19.10 Relations to Other Functions

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§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►

§19.10(ii) Elementary Functions

►If $y>0$ is assumed (without loss of generality), then …

48: 19.29 Reduction of General Elliptic Integrals

§19.29 Reduction of General Elliptic Integrals

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§19.29(i) Reduction Theorems

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§19.29(ii) Reduction to Basic Integrals

►(19.2.3) can be written … ►All other cases are integrals of the second kind. …

49: 2.6 Distributional Methods

… ►Although divergent, these integrals may be interpreted in a generalized sense. … ►Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►Also, …However, in the theory of generalized functions (distributions), there is a method, known as “regularization”, by which these integrals can be interpreted in a meaningful manner. …

50: 28.32 Mathematical Applications

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§28.32(i) Elliptical Coordinates and an Integral Relationship

►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. … … ► … ►The general paraboloidal coordinate system is linked with Cartesian coordinates via …