Neville’s

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1: 20.1 Special Notation

… ►Neville’s notation: ${\theta }_s(z|\tau )$, ${\theta }_c(z|\tau )$, ${\theta }_d(z|\tau )$, ${\theta }_n(z|\tau )$, respectively, for ${\theta }_3^2\left(0\right|\tau ){\theta }_1\left(u\right|\tau )/{\theta }_1^{\prime }\left(0\right|\tau )$, ${\theta }_2\left(u\right|\tau )/{\theta }_2\left(0\right|\tau )$, ${\theta }_3\left(u\right|\tau )/{\theta }_3\left(0\right|\tau )$, ${\theta }_4\left(u\right|\tau )/{\theta }_4\left(0\right|\tau )$, where again $u=z/{\theta }_3^2\left(0\right|\tau )$. …

2: 22.2 Definitions

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22.2.9 $$\mathrm{sc}(z,k)=\frac{{\theta }_3(0,q)}{{\theta }_4(0,q)}\frac{{\theta }_1(\zeta ,q)}{{\theta }_2(\zeta ,q)}=\frac{1}{\mathrm{cs}(z,k)}.$$

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Defines:

$\mathrm{cs}(z,k)$: Jacobian elliptic function and $\mathrm{sc}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/22.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

… ►The six functions containing the letter $\mathrm{s}$ in their two-letter name are odd in $z$; the other six are even in $z$. ►In terms of Neville’s theta functions (§20.1) …

3: 20.15 Tables

… ►Tables of Neville’s theta functions ${\theta }_s(x,q)$, ${\theta }_c(x,q)$, ${\theta }_d(x,q)$, ${\theta }_n(x,q)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\epsilon ,\alpha =0(5^{\circ }){90}^{\circ }$, where (in radian measure) $\epsilon =x/{\theta }_3^2(0,q)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …

4: 20.11 Generalizations and Analogs

… ►A further development on the lines of Neville’s notation (§20.1) is as follows. …

5: Bibliography N

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Modern Computing Methods. 2nd edition, Notes on Applied Science, No. 16, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (A. S. Householder), ZentralBlatt

Referenced by:

§3.8(iv).

Encodings:

BibTeX

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

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An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

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Jacobian Elliptic Functions. 2nd edition, Clarendon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§20.1, Notations T, Notations T, Notations T, Notations T.

Encodings:

BibTeX

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Combinatorial Algorithms. Academic Press, New York.

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External Links:

ISBN 0-12-519250-9, MathReview (Alon Itai), ZentralBlatt

Referenced by:

§26.22.

Encodings:

BibTeX

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6: 20.7 Identities

… ►Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices $2,3,4$. … ►

§20.7(v) Watson’s Identities

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