# complete

(0.001 seconds)

## 1—10 of 141 matching pages

##### 1: 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 $4\mathrm{i}{K^{\prime}}$. …
##### 2: 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 29.13.7: sE 5 m ⁡ ( x , 0.1 ) for - 2 ⁢ K ⁡ ≤ x ≤ 2 ⁢ K ⁡ , m = 0 , 1 , 2 . K ⁡ = 1.61244 ⁢ … . Magnify Figure 29.13.8: sE 5 m ⁡ ( x , 0.9 ) for - 2 ⁢ K ⁡ ≤ x ≤ 2 ⁢ K ⁡ , m = 0 , 1 , 2 . K ⁡ = 2.57809 ⁢ … . Magnify Figure 29.13.9: cE 5 m ⁡ ( x , 0.1 ) for - 2 ⁢ K ⁡ ≤ x ≤ 2 ⁢ K ⁡ , m = 0 , 1 , 2 . K ⁡ = 1.61244 ⁢ … . Magnify
##### 3: 22.3 Graphics Figure 22.3.16: sn ⁡ ( x + i ⁢ y , k ) for k = 0.99 , - 3 ⁢ K ⁡ ≤ x ≤ 3 ⁢ K ⁡ , 0 ≤ y ≤ 4 ⁢ K ′ ⁡ . K ⁡ = 3.3566 ⁢ … , K ′ ⁡ = 1.5786 ⁢ … . Magnify 3D Help Figure 22.3.17: cn ⁡ ( x + i ⁢ y , k ) for k = 0.99 , - 3 ⁢ K ⁡ ≤ x ≤ 3 ⁢ K ⁡ , 0 ≤ y ≤ 4 ⁢ K ′ ⁡ . K ⁡ = 3.3566 ⁢ … , K ′ ⁡ = 1.5786 ⁢ … . Magnify 3D Help Figure 22.3.18: dn ⁡ ( x + i ⁢ y , k ) for k = 0.99 , - 3 ⁢ K ⁡ ≤ x ≤ 3 ⁢ K ⁡ , 0 ≤ y ≤ 4 ⁢ K ′ ⁡ . K ⁡ = 3.3566 ⁢ … , K ′ ⁡ = 1.5786 ⁢ … . Magnify 3D Help Figure 22.3.19: cd ⁡ ( x + i ⁢ y , k ) for k = 0.99 , - 3 ⁢ K ⁡ ≤ x ≤ 3 ⁢ K ⁡ , 0 ≤ y ≤ 4 ⁢ K ′ ⁡ . K ⁡ = 3.3566 ⁢ … , K ′ ⁡ = 1.5786 ⁢ … . Magnify 3D Help Figure 22.3.20: dc ⁡ ( x + i ⁢ y , k ) for k = 0.99 , - 3 ⁢ K ⁡ ≤ x ≤ 3 ⁢ K ⁡ , 0 ≤ y ≤ 4 ⁢ K ′ ⁡ . K ⁡ = 3.3566 ⁢ … , K ′ ⁡ = 1.5786 ⁢ … . Magnify 3D Help
##### 4: 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. …
##### 5: 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. …
##### 6: 22.4 Periods, Poles, and Zeros
The other poles are at congruent points, which is the set of points obtained by making translations by $2mK+2niK^{\prime}$, where $m,n\in\mathbb{Z}$. … 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 $\mbox{s}=0$, $\mbox{c}=K$, $\mbox{d}=K+iK^{\prime}$, $\mbox{n}=iK^{\prime}$. 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 $m,n\in\mathbb{Z}$. … 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 $\operatorname{pq}\left(z,k\right)$. …
##### 7: 29.14 Orthogonality
is orthogonal and complete with respect to the inner product
29.14.2 $\langle g,h\rangle=\int_{0}^{K}\!\!\int_{0}^{{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \mathrm{d}t\mathrm{d}s,$
Each of the following seven systems is orthogonal and complete with respect to the inner product (29.14.2): …When combined, all eight systems (29.14.1) and (29.14.4)–(29.14.10) form an orthogonal and complete system with respect to the inner product
29.14.11 $\langle g,h\rangle=\int_{0}^{4K}\!\!\int_{0}^{2{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \mathrm{d}t\mathrm{d}s,$
##### 8: 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 $2K+\mathrm{i}{K^{\prime}}$. …
##### 9: 19.3 Graphics
See Figures 19.3.119.3.6 for complete and incomplete Legendre’s elliptic integrals. Figure 19.3.1: K ⁡ ( k ) and E ⁡ ( k ) as functions of k 2 for - 2 ≤ k 2 ≤ 1 . Graphs of K ′ ⁡ ( k ) and E ′ ⁡ ( k ) are the mirror images in the vertical line k 2 = 1 2 . Magnify Figure 19.3.5: Π ⁡ ( α 2 , k ) as a function of k 2 and α 2 for - 2 ≤ k 2 < 1 , - 2 ≤ α 2 ≤ 2 . …If α 2 = 0 , then it reduces to K ⁡ ( k ) . … Magnify 3D Help In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. …
##### 10: 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. … Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …