elliptic cases of R-a(b;z)

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1: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

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§23.2(iii) Periodicity

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2: 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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3: 22.2 Definitions

§22.2 Definitions

… ►where $K\left(k\right)$, $K^{\prime }\left(k\right)$ are defined in §19.2(ii). … ►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ► $\mathrm{s}\mathrm{s}(z,k)=1$. …

4: 19.16 Definitions

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§19.16(i) Symmetric Integrals

… ►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function … … ►

§19.16(iii) Various Cases of $R_{-a}(𝐛;𝐳)$

… ►All other elliptic cases are integrals of the second kind. …

5: 22.16 Related Functions

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

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Definition

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Relation to Elliptic Integrals

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Relation to the Elliptic Integral $E(\varphi ,k)$

… ►(Sometimes in the literature $\mathrm{Z}\left(x|k\right)$ is denoted by $\mathrm{Z}(\mathrm{am}(x,k),k^2)$.) …

6: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

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§19.2(ii) Legendre’s Integrals

… ►Also, if $k^2$ and ${\alpha }^2$ are real, then $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is called a circular or hyperbolic case according as ${\alpha }^2({\alpha }^2-k^2)({\alpha }^2-1)$ is negative or positive. The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points ${\alpha }^2=0,k^2,1$. … ►

§19.2(iii) Bulirsch’s Integrals

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7: 36.2 Catastrophes and Canonical Integrals

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Normal Forms for Umbilic Catastrophes with Codimension $K=3$

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Canonical Integrals

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§36.2(ii) Special Cases

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§36.2(iv) Addendum to 36.2(ii) Special Cases

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8: 22.21 Tables

§22.21 Tables

►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. ►Curtis (1964b) tabulates $\mathrm{sn}(mK/n,k)$, $\mathrm{cn}(mK/n,k)$, $\mathrm{dn}(mK/n,k)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. ►Lawden (1989, pp. 280–284 and 293–297) tabulates $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $ℰ(x,k)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. … ►Zhang and Jin (1996, p. 678) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. …

9: 23.23 Tables

§23.23 Tables

… ►2 in Abramowitz and Stegun (1964) gives values of $\mathrm{\wp }\left(z\right)$, ${\mathrm{\wp }}^{\prime }\left(z\right)$, and $\zeta \left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that ${\omega }_1=1$ and ${\omega }_3=\mathrm{i}a$ (rectangular case), or ${\omega }_1=1$ and ${\omega }_3=\frac{1}{2}+\mathrm{i}a$ (rhombic case), for $a$ = 1. …The values are tabulated on the real and imaginary $z$-axes, mostly ranging from 0 to 1 or $\mathrm{i}$ in steps of length 0. 05, and in the case of $\mathrm{\wp }\left(z\right)$ the user may deduce values for complex $z$ by application of the addition theorem (23.10.1). …

10: 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. …is independent of $z_1$, $z_2$, $z_3$. …