general elliptic functions

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11: 22.16 Related Functions

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22.16.1 $$\mathrm{am}(x,k)=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right),$$ $x\in ℝ$,

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Defines:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\in$: element of, $\mathrm{Arcsin}z$: general arcsine function, $ℝ$: real line, $x$: real and $k$: modulus

Referenced by:

§22.20(vi)

Permalink:

http://dlmf.nist.gov/22.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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12: 22.8 Addition Theorems

§22.8 Addition Theorems

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§22.8(iii) Special Relations Between Arguments

… ►If sums/differences of the $z_j$’s are rational multiples of $K\left(k\right)$, then further relations follow. …Generalizations are given in §22.9.

13: 12.15 Generalized Parabolic Cylinder Functions

§12.15 Generalized Parabolic Cylinder Functions

… ►can be viewed as a generalization of (12.2.4). This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

14: 19.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. … ►We use also the function $D(\varphi ,k)$, introduced by Jahnke et al. (1966, p. 43). The functions (19.1.1) and (19.1.2) are used in Erdélyi et al. (1953b, Chapter 13), except that $\mathrm{\Pi }({\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ are denoted by ${\mathrm{\Pi }}_1(\nu ,k)$ and $\mathrm{\Pi }(\varphi ,\nu ,k)$, respectively, where $\nu =-{\alpha }^2$. ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). … ► $R_{-a}(b_1,b_2,\mathrm{\dots },b_n;z_1,z_2,\mathrm{\dots },z_n)$ is a multivariate hypergeometric function that includes all the functions in (19.1.3). …

15: 36.1 Special Notation

… ►The main functions covered in this chapter are cuspoid catastrophes ${\mathrm{\Phi }}_K(t;𝐱)$; umbilic catastrophes with codimension three ${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)$, ${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)$; canonical integrals ${\mathrm{\Psi }}_K\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$; diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{E})}(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{H})}(𝐱;k)$ generated by the catastrophes. …

16: 22.9 Cyclic Identities

§22.9 Cyclic Identities

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§22.9(i) Notation

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17: 22.4 Periods, Poles, and Zeros

… ►The set of points $z=mK+n\mathrm{i}{K}^{\prime }$, $m,n\in \mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+n\mathrm{i}{K}^{\prime }$, where again $m,n\in \mathbb{Z}$. …

18: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

… ►The principal values satisfy … ►

§22.15(ii) Representations as Elliptic Integrals

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19: 19.17 Graphics

§19.17 Graphics

►See Figures 19.17.1–19.17.8 for symmetric elliptic integrals with real arguments. ►Because the $R$-function is homogeneous, there is no loss of generality in giving one variable the value $1$ or $-1$ (as in Figure 19.3.2). …The case $y=1$ corresponds to elementary functions. … ►

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Figure 19.17.8: $R_J(0,y,1,p)$, $0\le y\le 1$, $-1\le p\le 2$. … Magnify 3D Help

20: 32.10 Special Function Solutions

§32.10 Special Function Solutions

… ►More generally, if $n=1,2,3,\mathrm{\dots }$, then … ►Next, let $\mathrm{\Lambda }=\mathrm{\Lambda }(u,z)$ be the elliptic function (§§22.15(ii), 23.2(iii)) defined by …Then ${\text{P}}_{\text{VI}}$, with $\alpha =\beta =\gamma =0$ and $\delta =\frac{1}{2}$, has the general solution … ►