parametrization via Jacobian elliptic functions

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11: 5.15 Polygamma Functions

§5.15 Polygamma Functions

►The functions

\psi^{(n)}\left(z\right)

n=1,2,\dots

, are called the polygamma functions. In particular,

\psi'\left(z\right)

is the trigamma function;

\psi''

\psi^{(3)}

\psi^{(4)}

are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►For

B_{2k}

see §24.2(i). …

12: 9.12 Scorer Functions

§9.12 Scorer Functions

… ►where … ►

§9.12(ii) Graphs

… ► … ►

Functions and Derivatives

…

13: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

►

§14.19(i) Introduction

… ►

§14.19(ii) Hypergeometric Representations

… ►

§14.19(iv) Sums

… ►

§14.19(v) Whipple’s Formula for Toroidal Functions

…

14: 11.9 Lommel Functions

§11.9 Lommel Functions

… ► ►

Reflection Formulas

… ►

§11.9(ii) Expansions in Series of Bessel Functions

… ►

15: 20.2 Definitions and Periodic Properties

… ►

§20.2(i) Fourier Series

… ►

§20.2(ii) Periodicity and Quasi-Periodicity

… ►The theta functions are quasi-periodic on the lattice: … ►

§20.2(iii) Translation of the Argument by Half-Periods

… ►

§20.2(iv) $z$ -Zeros

…

16: 16.13 Appell Functions

§16.13 Appell Functions

►The following four functions of two real or complex variables

x

and

y

cannot be expressed as a product of two

{{}_{2}F_{1}}

functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): ►

16.13.1

{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m}}{\left(\beta^{% \prime}\right)_{n}}}{{\left(\gamma\right)_{m+n}}m!n!}x^{m}y^{n},

\max\left(|x|,|y|\right)<1

ⓘ

Defines:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.13 and Ch.16

… ►

16.13.4

{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m+n}}}{{\left(% \gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n}}m!n!}x^{m}y^{n},

\sqrt{|x|}+\sqrt{|y|}<1

ⓘ

Defines:: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/16.13.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.13 and Ch.16

… ► …

17: 15.2 Definitions and Analytical Properties

… ►

§15.2(i) Gauss Series

►The hypergeometric function

F\left(a,b;c;z\right)

is defined by the Gauss series … … ►On the circle of convergence,

|z|=1

, the Gauss series: … ►

§15.2(ii) Analytic Properties

…

18: 22.16 Related Functions

… ►

§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function

… ►

Relation to Elliptic Integrals

… ►

§22.16(ii) Jacobi’s Epsilon Function

… ►See Figure 22.16.2. … ►

§22.16(iii) Jacobi’s Zeta Function

…

19: 10.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►For the spherical Bessel functions and modified spherical Bessel functions the order

n

is a nonnegative integer. For the other functions when the order

\nu

is replaced by

n

, it can be any integer. For the Kelvin functions the order

\nu

is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

20: 4.2 Definitions