Jacobi elliptic form

SearchAdvancedHelp

(0.003 seconds)

11—15 of 15 matching pages

11: Bibliography K

… ►

Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt

Referenced by:

§22.9(iv), §22.9(v).

Encodings:

BibTeX

►

Cyclic identities involving Jacobi elliptic functions. J. Math. Phys. 43 (7), pp. 3798–3806.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§22.9(i), §22.9(ii), §22.9(iii), §22.9(iv), §22.9(v).

Encodings:

BibTeX

►

Connecting Jacobi elliptic functions with different modulus parameters. Pramana 63 (5), pp. 921–936.

ⓘ

External Links:

Pdf, Document

Referenced by:

§22.7(iii).

Encodings:

BibTeX

… ►

A vortex filament moving without change of form. J. Fluid Mech. 112, pp. 397–409.

ⓘ

External Links:

ISSN 0022-1120, Document, MathReview, ZentralBlatt

Referenced by:

§19.35(ii).

Encodings:

BibTeX

… ►

Introduction to Elliptic Curves and Modular Forms. 2nd edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-97966-2, MathReview, ZentralBlatt

Referenced by:

§20.10(i), §20.12(i), §20.9(iii), §22.18(iv), §23.1, §23.20(ii), §23.20(v), Ch.23.

Encodings:

BibTeX

…

12: 23.15 Definitions

… ►The set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)). … ►If, as a function of $q$, $f(\tau )$ is analytic at $q=0$, then $f(\tau )$ is called a modular form. If, in addition, $f(\tau )\to 0$ as $q\to 0$, then $f(\tau )$ is called a cusp form. … ►

Elliptic Modular Function

…

13: 29.14 Orthogonality

… ►

29.14.2 $$⟨g,h⟩={\int }_0^K{\int }_0^{K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$

ⓘ

Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $k$: real parameter and $w(s,t)$: function

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

… ►

29.14.3 $$w(s,t)={\mathrm{sn}}^2(K+\mathrm{i}t,k)-{\mathrm{sn}}^2(s,k).$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $k$: real parameter and $w(s,t)$: function

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

… ►

29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

… ►When combined, all eight systems (29.14.1) and (29.14.4)–(29.14.10) form an orthogonal and complete system with respect to the inner product ►

29.14.11 $$⟨g,h⟩={\int }_0^{4K}{\int }_0^{2K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$

ⓘ

Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $k$: real parameter and $w(s,t)$: function

Permalink:

http://dlmf.nist.gov/29.14.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

…

14: 22.9 Cyclic Identities

§22.9 Cyclic Identities

►

§22.9(i) Notation

… ► … ► … ►

15: Bibliography L

… ►

Reduction of Elliptic Integrals to Legendre Normal Form. Technical report Technical Report 97-21, Department of Computer Science, University of Waterloo, Waterloo, Ontario.

ⓘ

Referenced by:

§19.14(ii).

Encodings:

BibTeX

… ►

Co-recursive associated Jacobi polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 203–213.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

… ►

Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

BibTeX

… ►

Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

Encodings:

BibTeX

… ►

Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.

ⓘ

External Links:

FullText, MathReview (J. Bryant), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

…