general elliptic functions

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31: Bibliography R

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Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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External Links:

MathReview (H. H. Martens), ZentralBlatt

Referenced by:

§20.11(iv), §20.12(ii).

Encodings:

BibTeX

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Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

Encodings:

BibTeX

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Total positivity properties of generalized hypergeometric functions of matrix argument. J. Statist. Phys. 116 (1-4), pp. 907–922.

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External Links:

ISSN 0022-4715, Document, MathReview (Stamatis Koumandos), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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External Links:

ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§20.11(iii).

Encodings:

BibTeX

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Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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External Links:

ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

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32: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

►Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s $F_D$ (Carlson (1961b)). … ► … ►Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. … ►

33: 32.2 Differential Equations

… ►In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. … ►The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$. ►For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. … ►If $\delta \ne 0$ in ${\text{P}}_{\text{V}}$, then set $\delta =-\frac{1}{2}$, without loss of generality. … ►

§32.2(iv) Elliptic Form

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34: 23.20 Mathematical Applications

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§23.20(i) Conformal Mappings

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§23.20(ii) Elliptic Curves

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§23.20(iii) Factorization

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§23.20(v) Modular Functions and Number Theory

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35: 19.20 Special Cases

§19.20 Special Cases

… ►The general lemniscatic case is … ►where $x,y,z$ may be permuted. … ►The general lemniscatic case is … ►

36: Bibliography

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A general mode theory for the elliptic disk microstrip antenna. IEEE Trans. Antennas and Propagation 43 (6), pp. 560–568.

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External Links:

Document

Referenced by:

5th item.

Encodings:

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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

Encodings:

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Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.

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External Links:

ISSN 0030-8730, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.35(i).

Encodings:

BibTeX

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Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23 (2), pp. 512–524.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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Hypergeometric Functions and Elliptic Integrals. In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.), pp. 48–85.

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External Links:

MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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37: 23.15 Definitions

§23.15 Definitions

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§23.15(i) General Modular Functions

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Elliptic Modular Function

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Dedekind’s Eta Function (or Dedekind Modular Function)

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38: 36.10 Differential Equations

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36.10.3 $$\frac{{\partial }^2{\mathrm{\Psi }}_1}{{\partial x}^2}-\frac{x}{3}{\mathrm{\Psi }}_1=0.$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

… ►In terms of the normal forms (36.2.2) and (36.2.3), the ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$ satisfy the following operator equations … ►

36.10.13 $$6\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x\partial y}-2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial y}+y{\mathrm{\Psi }}^{(\mathrm{E})}=0,$$

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Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iii), §36.10 and Ch.36

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36.10.14 $$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0.$$

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Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

Erratum (V1.0.1) for Equation (36.10.14)

Permalink:

http://dlmf.nist.gov/36.10.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally appeared with $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}$, rather than $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}$:

$$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0.$$

See also:

Annotations for §36.10(iii), §36.10 and Ch.36

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36.10.17 $$\mathrm{i}\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial z}=\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}+\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2},$$

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Symbols:

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter, $z$: real parameter and $x$: real parameter

Referenced by:

§36.10(iv)

Permalink:

http://dlmf.nist.gov/36.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(iv), §36.10 and Ch.36

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39: 19.11 Addition Theorems

§19.11 Addition Theorems

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§19.11(i) General Formulas

… ►Hence, care has to be taken with the multivalued functions in (19.11.5). … ►

19.11.8 $$E(\varphi ,k)=E\left(k\right)-E(\theta ,k)+k^2\sin \theta \sin \varphi ,$$

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Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

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§19.11(iii) Duplication Formulas

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40: 23 Weierstrass Elliptic and Modular
Functions

Chapter 23 Weierstrass Elliptic and Modular Functions

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