# elliptic integrals

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## 11—20 of 131 matching pages

##### 11: 22.4 Periods, Poles, and Zeros
Figure 22.4.1 illustrates the locations in the $z$-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices $0$, $2K$, $2K+2iK^{\prime}$, $2iK^{\prime}$. … Figure 22.4.2 depicts the fundamental unit cell in the $z$-plane, with vertices $\mbox{s}=0$, $\mbox{c}=K$, $\mbox{d}=K+iK^{\prime}$, $\mbox{n}=iK^{\prime}$. … This half-period will be plus or minus a member of the triple ${K,iK^{\prime},K+iK^{\prime}}$; the other two members of this triple are quarter periods of $\operatorname{pq}\left(z,k\right)$. …
##### 12: 19.13 Integrals of Elliptic Integrals
###### §19.13(i) Integration with Respect to the Modulus
Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$, together with special cases.
###### §19.13(iii) Laplace Transforms
For direct and inverse Laplace transforms for the complete elliptic integrals $K\left(k\right)$, $E\left(k\right)$, and $D\left(k\right)$ see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.
##### 13: 29.17 Other Solutions
29.17.1 $F(z)=E(z)\int_{\mathrm{i}{K^{\prime}}}^{z}\frac{\mathrm{d}u}{(E(u))^{2}}.$
They are algebraic functions of $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, and $\operatorname{dn}\left(z,k\right)$, and have primitive period $8K$. … Lamé–Wangerin functions are solutions of (29.2.1) with the property that $(\operatorname{sn}\left(z,k\right))^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K^{\prime}}$ to $2K+\mathrm{i}{K^{\prime}}$. …
##### 14: 22.11 Fourier and Hyperbolic Series
22.11.3 $\operatorname{dn}\left(z,k\right)=\frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{% \infty}\frac{q^{n}\cos\left(2n\zeta\right)}{1+q^{2n}}.$
Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and $q\exp\left(2|\Im\zeta|\right)<1$,
22.11.13 ${\operatorname{sn}}^{2}\left(z,k\right)=\frac{1}{k^{2}}\left(1-\frac{E}{K}% \right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}% \cos\left(2n\zeta\right).$
22.11.14 $k^{2}{\operatorname{sn}}^{2}\left(z,k\right)=\frac{{E^{\prime}}}{{K^{\prime}}}% -\left(\frac{\pi}{2{K^{\prime}}}\right)^{2}\sum_{n=-\infty}^{\infty}\left({% \operatorname{sech}}^{2}\left(\frac{\pi}{2{K^{\prime}}}(z-2nK)\right)\right),$
where ${E^{\prime}}={E^{\prime}}\left(k\right)$ is defined by §19.2.9. …
##### 16: 19.38 Approximations
###### §19.38 Approximations
Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^{2}$ with $0\leq m<1$ can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for $0 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …
##### 17: 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 $E\left(k\right)$. …
##### 18: 29.14 Orthogonality
29.14.2 $\langle g,h\rangle=\int_{0}^{K}\!\!\int_{0}^{{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \mathrm{d}t\mathrm{d}s,$
29.14.3 $w(s,t)={\operatorname{sn}}^{2}\left(K+\mathrm{i}t,k\right)-{\operatorname{sn}}% ^{2}\left(s,k\right).$
29.14.4 $\mathit{sE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{sE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
29.14.5 $\mathit{cE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{cE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
29.14.11 $\langle g,h\rangle=\int_{0}^{4K}\!\!\int_{0}^{2{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \mathrm{d}t\mathrm{d}s,$
##### 19: 29.16 Asymptotic Expansions
The approximations for Lamé polynomials hold uniformly on the rectangle $0\leq\Re z\leq K$, $0\leq\Im z\leq{K^{\prime}}$, when $nk$ and $nk^{\prime}$ assume large real values. …
##### 20: 19.1 Special Notation
 $l,m,n$ nonnegative integers. …
$K\left(k\right),$
$E\left(k\right),$
$F\left(\phi,k\right),$
$E\left(\phi,k\right),$