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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 29.13.21: | 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
##### 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: 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. …
##### 5: 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
##### 6: 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 $\operatorname{pq}\left(z,k\right)$. …
##### 7: 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 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
##### 8: 14.10 Recurrence Relations and Derivatives
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 ${(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)-(2\nu+3)x\mathsf{P}^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0,$
14.10.5 $\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}=(\nu+\mu)\mathsf{P}^{\mu}_{\nu-1}\left(x\right)-\nu x\mathsf{P}^{% \mu}_{\nu}\left(x\right).$
$\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ 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).
##### 9: 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 $2K+\mathrm{i}{K^{\prime}}$. …
##### 10: 14.33 Tables
• Abramowitz and Stegun (1964, Chapter 8) tabulates $\mathsf{P}^{n}\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 5–8D; $\mathsf{P}^{n}'\left(x\right)$ for $n=1(1)4,9,10$, $x=0(.01)1$, 5–7D; $\mathsf{Q}^{n}\left(x\right)$ and $\mathsf{Q}^{n}'\left(x\right)$ for $n=0(1)3,9,10$, $x=0(.01)1$, 6–8D; $P^{n}\left(x\right)$ and $P^{n}'\left(x\right)$ for $n=0(1)5,9,10$, $x=1(.2)10$, 6S; $Q^{n}\left(x\right)$ and $Q^{n}'\left(x\right)$ for $n=0(1)3,9,10$, $x=1(.2)10$, 6S. (Here primes denote derivatives with respect to $x$.)

• Zhang and Jin (1996, Chapter 4) tabulates $\mathsf{P}^{n}\left(x\right)$ for $n=2(1)5,10$, $x=0(.1)1$, 7D; $\mathsf{P}^{n}\left(\cos\theta\right)$ for $n=1(1)4,10$, $\theta=0(5^{\circ})90^{\circ}$, 8D; $\mathsf{Q}^{n}\left(x\right)$ for $n=0(1)2,10$, $x=0(.1)0.9$, 8S; $\mathsf{Q}^{n}\left(\cos\theta\right)$ for $n=0(1)3,10$, $\theta=0(5^{\circ})90^{\circ}$, 8D; $\mathsf{P}^{m}_{n}\left(x\right)$ for $m=1(1)4$, $n-m=0(1)2$, $n=10$, $x=0,0.5$, 8S; $\mathsf{Q}^{m}_{n}\left(x\right)$ for $m=1(1)4$, $n=0(1)2,10$, 8S; $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ for $m=0(1)3$, $\nu=0(.25)5$, $\theta=0(15^{\circ})90^{\circ}$, 5D; $P^{n}\left(x\right)$ for $n=2(1)5,10$, $x=1(1)10$, 7S; $Q^{n}\left(x\right)$ for $n=0(1)2,10$, $x=2(1)10$, 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 $\nu$-zeros of $\mathsf{P}^{m}_{\nu}\left(\cos\theta\right)$ and of its derivative for $m=0(1)4$, $\theta=10^{\circ},30^{\circ},150^{\circ}$.

• Belousov (1962) tabulates $\mathsf{P}^{m}_{n}\left(\cos\theta\right)$ (normalized) for $m=0(1)36$, $n-m=0(1)56$, $\theta=0(2.5^{\circ})90^{\circ}$, 6D.

• Žurina and Karmazina (1964, 1965) tabulate the conical functions $\mathsf{P}^{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)50$, $x=-0.9(.1)0.9$, 7S; $P^{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)50$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when $-1.

• Žurina and Karmazina (1963) tabulates the conical functions $\mathsf{P}^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)25$, $x=-0.9(.1)0.9$, 7S; $P^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)25$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when $-1.