# general elliptic functions

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## 11—20 of 91 matching pages

##### 11: 22.16 Related Functions
22.16.1 $\operatorname{am}\left(x,k\right)=\operatorname{Arcsin}\left(\operatorname{sn}% \left(x,k\right)\right),$ $x\in\mathbb{R}$,
##### 12: 22.8 Addition Theorems
###### §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 $D\left(\phi,k\right)$, 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 $\Pi\left(\alpha^{2},k\right)$ and $\Pi\left(\phi,\alpha^{2},k\right)$ are denoted by $\Pi_{1}(\nu,k)$ and $\Pi(\phi,\nu,k)$, 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 $K(\alpha)$, $E(\alpha)$, $\Pi(n\backslash\alpha)$, $F(\phi\backslash\alpha)$, $E(\phi\backslash\alpha)$, and $\Pi(n;\phi\backslash\alpha)$, where $\alpha=\operatorname{arcsin}k$ and $n$ is the $\alpha^{2}$ (not related to $k$) in (19.1.1) and (19.1.2). … $R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)$ 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 $\Phi_{K}\left(t;\mathbf{x}\right)$; umbilic catastrophes with codimension three $\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)$, $\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)$; canonical integrals $\Psi_{K}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)$; diffraction catastrophes $\Psi_{K}(\mathbf{x};k)$, $\Psi^{(\mathrm{E})}(\mathbf{x};k)$, $\Psi^{(\mathrm{H})}(\mathbf{x};k)$ generated by the catastrophes. …
##### 17: 22.4 Periods, Poles, and Zeros
The set of points $z=mK+niK^{\prime}$, $m,n\in\mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+niK^{\prime}$, where again $m,n\in\mathbb{Z}$. …
##### 18: 22.15 Inverse Functions
###### §22.15(i) Definitions
The principal values satisfy …
##### 19: 19.17 Graphics
###### §19.17 Graphics
See Figures 19.17.119.17.8 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. … Figure 19.17.8: R J ⁡ ( 0 , y , 1 , p ) , 0 ≤ y ≤ 1 , - 1 ≤ p ≤ 2 . … Magnify 3D Help
##### 20: 32.10 Special Function Solutions
###### §32.10 Special Function Solutions
More generally, if $n=1,2,3,\dots$, then … Next, let $\Lambda=\Lambda(u,z)$ be the elliptic function (§§22.15(ii), 23.2(iii)) defined by …Then $\mbox{P}_{\mbox{\scriptsize VI}}$, with $\alpha=\beta=\gamma=0$ and $\delta=\tfrac{1}{2}$, has the general solution …