# elliptical

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##### 2: 22.15 Inverse Functions
###### §22.15(i) Definitions
The principal values satisfy …
##### 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. … … $\operatorname{ss}\left(z,k\right)=1$. …
##### 4: 19.16 Definitions
###### §19.16(i) Symmetric Integrals
The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … …
###### §19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$
All other elliptic cases are integrals of the second kind. …
##### 6: 19.2 Definitions
###### §19.2(i) General Elliptic Integrals
is called an elliptic integral. …Thus the elliptic part of (19.2.1) is …
##### 7: 36.2 Catastrophes and Canonical Integrals
###### Normal Forms for Umbilic Catastrophes with Codimension $K=3$
36.2.2 $\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)=s^{3}-3st^{2}+z(s^{2}+t^{2})+yt% +xs,$ $\mathbf{x}=\{x,y,z\}$,
(elliptic umbilic). …
##### 8: 19.35 Other Applications
###### §19.35(ii) Physical
Elliptic integrals appear in lattice models of critical phenomena (Guttmann and Prellberg (1993)); theories of layered materials (Parkinson (1969)); fluid dynamics (Kida (1981)); string theory (Arutyunov and Staudacher (2004)); astrophysics (Dexter and Agol (2009)).
22.8.14 $\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{cn% }u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$
If sums/differences of the $z_{j}$’s are rational multiples of $K\left(k\right)$, then further relations follow. …
For example, $\operatorname{sn}\left(z+K,k\right)=\operatorname{cd}\left(z,k\right)$. …