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##### 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: 10.35 Generating Function and Associated Series
10.35.4 $1=I_{0}\left(z\right)-2I_{2}\left(z\right)+2I_{4}\left(z\right)-2I_{6}\left(z% \right)+\dotsb,$
10.35.5 $e^{\pm z}=I_{0}\left(z\right)\pm 2I_{1}\left(z\right)+2I_{2}\left(z\right)\pm 2% I_{3}\left(z\right)+\dotsb,$
$\cosh z=I_{0}\left(z\right)+2I_{2}\left(z\right)+2I_{4}\left(z\right)+2I_{6}% \left(z\right)+\dots,$
$\sinh z=2I_{1}\left(z\right)+2I_{3}\left(z\right)+2I_{5}\left(z\right)+\dots.$
##### 4: 29.14 Orthogonality
First, the orthogonality relations (29.3.19) apply; see §29.12(i). …
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 $\mathit{sE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{sE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
29.14.11 $\langle g,h\rangle=\int_{0}^{4K}\!\!\int_{0}^{2{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \,\mathrm{d}t\,\mathrm{d}s,$
##### 5: 14.10 Recurrence Relations and Derivatives
14.10.1 ${\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}% \mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{% \nu}\left(x\right)=0,$
14.10.2 ${\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)% \mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)x\mathsf{P}^{\mu}_{\nu}% \left(x\right)=0,$
14.10.3 ${(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)-(2\nu+3)x\mathsf{P}^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0,$
$\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ also satisfies (14.10.1)–(14.10.5). …In addition, $P^{\mu}_{\nu}\left(x\right)$ and $Q^{\mu}_{\nu}\left(x\right)$ satisfy (14.10.3)–(14.10.5).
##### 6: 10.28 Wronskians and Cross-Products
10.28.1 $\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}\left(z\right)I_{-\nu}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
10.28.2 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
##### 7: 10.33 Continued Fractions
Assume $I_{\nu-1}\left(z\right)\neq 0$. …
10.33.1 $\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{1}{2\nu z^{-1}+}% \cfrac{1}{2(\nu+1)z^{-1}+}\cfrac{1}{2(\nu+2)z^{-1}+}\cdots,$ $z\neq 0$,
10.33.2 $\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{\frac{1}{2}z/\nu}% {1+}\cfrac{\frac{1}{4}z^{2}/(\nu(\nu+1))}{1+}\cfrac{\frac{1}{4}z^{2}/((\nu+1)(% \nu+2))}{1+}\cdots,$ $\nu\neq 0,-1,-2,\dotsc$.
##### 8: 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
##### 9: 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. …
##### 10: 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 $\operatorname{pq}\left(z,k\right)$. …