# elliptic integrals

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##### 1: 19.16 Definitions
###### §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 …The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … …
##### 2: 19.2 Definitions
###### §19.2(i) General EllipticIntegrals
is called an elliptic integral. …
###### §19.2(ii) Legendre’s Integrals
${K^{\prime}}\left(k\right)=K\left(k^{\prime}\right),$
##### 3: 36.2 Catastrophes and Canonical Integrals
###### Canonical Integrals
$\Psi^{(\mathrm{E})}\left(\boldsymbol{{0}}\right)=\tfrac{1}{3}\sqrt{\pi}\Gamma% \left(\tfrac{1}{6}\right)=3.28868,$
36.2.26 $\Psi^{(\mathrm{E})}\left(-\tfrac{1}{2}x\mp\tfrac{\sqrt{3}}{2}y,\pm\tfrac{\sqrt% {3}}{2}x-\tfrac{1}{2}y,z\right)=\Psi^{(\mathrm{E})}\left(x,y,z\right),$
36.2.28 $\Psi^{(\mathrm{E})}\left(0,0,z\right)=\overline{\Psi^{(\mathrm{E})}\left(0,0,-% z\right)}\\ =2\pi\sqrt{\frac{\pi z}{27}}\exp\left(\frac{2}{27}iz^{3}\right)\*\left(J_{-1/6% }\left(\frac{2}{27}z^{3}\right)+iJ_{1/6}\left(\frac{2}{27}z^{3}\right)\right),$ $z\geq 0$,
##### 4: 29.10 Lamé Functions with Imaginary Periods
29.10.2 $z^{\prime}=\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),$
$\mathit{Ec}^{2m}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{\prime}% }^{2}\right),$
$\mathit{Ec}^{2m+1}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$
$\mathit{Es}^{2m+1}_{\nu}\left(\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),{k^{% \prime}}^{2}\right),$
The first and the fourth functions have period $2\mathrm{i}{K^{\prime}}$; the second and the third have period $4\mathrm{i}{K^{\prime}}$. …
##### 5: 29.13 Graphics Figure 29.13.5: uE 4 m ⁡ ( x , 0.1 ) for - 2 ⁢ K ⁡ ≤ x ≤ 2 ⁢ K ⁡ , m = 0 , 1 , 2 . K ⁡ = 1.61244 ⁢ … . Magnify Figure 29.13.6: uE 4 m ⁡ ( x , 0.9 ) for - 2 ⁢ K ⁡ ≤ x ≤ 2 ⁢ K ⁡ , m = 0 , 1 , 2 . K ⁡ = 2.57809 ⁢ … . Magnify Figure 29.13.21: | uE 4 1 ⁡ ( x + i ⁢ y , 0.1 ) | for - 3 ⁢ K ⁡ ≤ x ≤ 3 ⁢ K ⁡ , 0 ≤ y ≤ 2 ⁢ K ′ ⁡ . K ⁡ = 1.61244 ⁢ … , K ′ ⁡ = 2.57809 ⁢ … . Magnify 3D Help Figure 29.13.22: | uE 4 1 ⁡ ( x + i ⁢ y , 0.5 ) | for - 3 ⁢ K ⁡ ≤ x ≤ 3 ⁢ K ⁡ , 0 ≤ y ≤ 2 ⁢ K ′ ⁡ . K ⁡ = K ′ ⁡ = 1.85407 ⁢ … . Magnify 3D Help Figure 29.13.23: | uE 4 1 ⁡ ( x + i ⁢ y , 0.9 ) | for - 3 ⁢ K ⁡ ≤ x ≤ 3 ⁢ K ⁡ , 0 ≤ y ≤ 2 ⁢ K ′ ⁡ . K ⁡ = 2.57809 ⁢ … , K ′ ⁡ = 1.61244 ⁢ … . Magnify 3D Help
##### 6: 22.21 Tables
Spenceley and Spenceley (1947) tabulates $\operatorname{sn}\left(Kx,k\right)$, $\operatorname{cn}\left(Kx,k\right)$, $\operatorname{dn}\left(Kx,k\right)$, $\operatorname{am}\left(Kx,k\right)$, $\mathcal{E}\left(Kx,k\right)$ for $\operatorname{arcsin}k=1^{\circ}(1^{\circ})89^{\circ}$ and $x=0\left(\tfrac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\operatorname{am}\left(Kx,k\right)$. Curtis (1964b) tabulates $\operatorname{sn}\left(mK/n,k\right)$, $\operatorname{cn}\left(mK/n,k\right)$, $\operatorname{dn}\left(mK/n,k\right)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. … Zhang and Jin (1996, p. 678) tabulates $\operatorname{sn}\left(Kx,k\right)$, $\operatorname{cn}\left(Kx,k\right)$, $\operatorname{dn}\left(Kx,k\right)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. …
##### 7: 19.4 Derivatives and Differential Equations
###### §19.4(i) Derivatives
$\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}{\mathrm{d}k% }=kK\left(k\right),$
$\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-K\left(k% \right)}{k},$
$\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}{\mathrm{d}k}=-\frac{kE\left% (k\right)}{{k^{\prime}}^{2}},$
If $\phi=\pi/2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. …
##### 8: 19.6 Special Cases
###### §19.6(i) Complete EllipticIntegrals
$K\left(0\right)=E\left(0\right)={K^{\prime}}\left(1\right)={E^{\prime}}\left(1% \right)=\tfrac{1}{2}\pi,$
$K\left(1\right)={K^{\prime}}\left(0\right)=\infty,$
##### 9: 19.7 Connection Formulas
19.7.1 $E\left(k\right){K^{\prime}}\left(k\right)+{E^{\prime}}\left(k\right)K\left(k% \right)-K\left(k\right){K^{\prime}}\left(k\right)=\tfrac{1}{2}\pi.$
$K\left(1/k\right)=k(K\left(k\right)\mp\mathrm{i}K\left(k^{\prime}\right)),$
$K\left(1/k^{\prime}\right)=k^{\prime}(K\left(k^{\prime}\right)\pm\mathrm{i}K% \left(k\right)),$
$E\left(\phi,k_{1}\right)=(E\left(\beta,k\right)-{k^{\prime}}^{2}F\left(\beta,k% \right))/k,$
##### 10: 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)).