Jacobian normal form

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1: 22.18 Mathematical Applications

… ►The special case $y^2=(1-x^2)(1-k^2{x}^2)$ is in Jacobian normal form. For any two points $(x_1,y_1)$ and $(x_2,y_2)$ on this curve, their sum $(x_3,y_3)$, always a third point on the curve, is defined by the Jacobi–Abel addition law …

2: 22.15 Inverse Functions

… ►The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. …

3: Bibliography C

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Normal elliptic integrals of the first and second kinds. Duke Math. J. 31 (3), pp. 405–419.

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External Links:

Document, MathReview (S. Hitotumatu), ZentralBlatt

Referenced by:

§19.22(iii), §19.23, §19.33(i), §19.33(i).

Encodings:

BibTeX

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Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§19.25(v), §19.25(v), §22.13(i), §22.6(ii), §22.6(iii), §22.8(i), §22.8(ii), Profile Bille C. Carlson.

Encodings:

BibTeX

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Jacobian elliptic functions as inverses of an integral. J. Comput. Appl. Math. 174 (2), pp. 355–359.

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External Links:

ISSN 0377-0427, Document, MathReview (R. N. Kalia), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

Encodings:

BibTeX

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Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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External Links:

ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

Encodings:

BibTeX

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Algorithm AS 24: From normal integral to deviate. Appl. Statist. 18 (3), pp. 290–293.

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Notes:

Includes Fortran program, maximum accuracy 12D.

External Links:

FullText

Referenced by:

§7.25(ii).

Encodings:

BibTeX

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4: 29.15 Fourier Series and Chebyshev Series

… ►be the eigenvector corresponding to $H_m$ and normalized so that … ►Since (29.2.5) implies that $\cos \varphi =\mathrm{sn}(z,k)$, (29.15.1) can be rewritten in the form …The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an $(n+1)\times (n+1)$ tridiagonal matrix; see Arscott and Khabaza (1962). … ►

29.15.49 $${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+1}{U}_{2p}\left(\mathrm{sn}(z,k)\right),$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.50 $${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+2}{U}_{2p+1}\left(\mathrm{sn}(z,k)\right).$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $D_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

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5: 31.2 Differential Equations

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§31.2(ii) Normal Form of Heun’s Equation

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§31.2(iii) Trigonometric Form

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§31.2(iv) Doubly-Periodic Forms

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Jacobi’s Elliptic Form

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Weierstrass’s Form

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