# Jacobi

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##### 2: 18.3 Definitions
###### §18.3 Definitions
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … The Jacobi polynomials are in powers of $x-1$ for $n=0,1,\dots,6$. … However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). …
22.8.3 $\operatorname{dn}(u+v)=\frac{\operatorname{dn}u\operatorname{dn}v-k^{2}% \operatorname{sn}u\operatorname{cn}u\operatorname{sn}v\operatorname{cn}v}{1-k^% {2}{\operatorname{sn}^{2}}u{\operatorname{sn}^{2}}v}.$
22.8.14 $\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{cn% }u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$
22.8.23 $\begin{vmatrix}\operatorname{sn}z_{1}\operatorname{cn}z_{1}&\operatorname{cn}z% _{1}\operatorname{dn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}z_{1}\\ \operatorname{sn}z_{2}\operatorname{cn}z_{2}&\operatorname{cn}z_{2}% \operatorname{dn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}\\ \operatorname{sn}z_{3}\operatorname{cn}z_{3}&\operatorname{cn}z_{3}% \operatorname{dn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}\\ \operatorname{sn}z_{4}\operatorname{cn}z_{4}&\operatorname{cn}z_{4}% \operatorname{dn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}\end{% vmatrix}=0.$
##### 4: 22.6 Elementary Identities
22.6.2 $1+{\operatorname{cs}^{2}}\left(z,k\right)=k^{2}+{\operatorname{ds}^{2}}\left(z% ,k\right)={\operatorname{ns}^{2}}\left(z,k\right),$
22.6.5 $\operatorname{sn}\left(2z,k\right)=\frac{2\operatorname{sn}\left(z,k\right)% \operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)}{1-k^{2}{% \operatorname{sn}^{4}}\left(z,k\right)},$
22.6.8 $\operatorname{cd}\left(2z,k\right)=\frac{{\operatorname{cd}^{2}}\left(z,k% \right)-{k^{\prime}}^{2}{\operatorname{sd}^{2}}\left(z,k\right){\operatorname{% nd}^{2}}\left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{4}}\left(% z,k\right)},$
##### 5: 22.21 Tables
###### §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 $\operatorname{am}\left(Kx,k\right)$. … Lawden (1989, pp. 280–284 and 293–297) tabulates $\operatorname{sn}\left(x,k\right)$, $\operatorname{cn}\left(x,k\right)$, $\operatorname{dn}\left(x,k\right)$, $\mathcal{E}\left(x,k\right)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. … 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
For example, the poles of $\operatorname{sn}\left(z,k\right)$, abbreviated as $\operatorname{sn}$ in the following tables, are at $z=2mK+(2n+1)iK^{\prime}$. … Then: (a) In any lattice unit cell $\operatorname{pq}\left(z,k\right)$ has a simple zero at $z=\mbox{p}$ and a simple pole at $z=\mbox{q}$. (b) The difference between p and the nearest q is a half-period of $\operatorname{pq}\left(z,k\right)$. 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 $\operatorname{pq}\left(z,k\right)$. … For example, $\operatorname{sn}\left(z+K,k\right)=\operatorname{cd}\left(z,k\right)$. …
##### 7: 20.4 Values at $z$ = 0
20.4.1 $\theta_{1}\left(0,q\right)=\theta_{2}'\left(0,q\right)=\theta_{3}'\left(0,q% \right)=\theta_{4}'\left(0,q\right)=0,$
###### Jacobi’s Identity
20.4.6 $\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(0,q\right).$
20.4.7 $\theta_{1}''(0,q)=\theta_{2}'''\left(0,q\right)=\theta_{3}'''\left(0,q\right)=% \theta_{4}'''\left(0,q\right)=0.$
20.4.12 $\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'\left(0,q\right)}=\frac{\theta% _{2}''\left(0,q\right)}{\theta_{2}\left(0,q\right)}+\frac{\theta_{3}''\left(0,% q\right)}{\theta_{3}\left(0,q\right)}+\frac{\theta_{4}''\left(0,q\right)}{% \theta_{4}\left(0,q\right)}.$
##### 8: 22.13 Derivatives and Differential Equations
22.13.1 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sn}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{sn}^{2}}\left(z,k\right)\right)\left(1-k^{2}{% \operatorname{sn}^{2}}\left(z,k\right)\right),$
22.13.2 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cn}\left(z,k\right)\right)^{% 2}={\left(1-{\operatorname{cn}^{2}}\left(z,k\right)\right)}{\left({k^{\prime}}% ^{2}+k^{2}{\operatorname{cn}^{2}}\left(z,k\right)\right)},$
22.13.3 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dn}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{dn}^{2}}\left(z,k\right)\right)\left({\operatorname{% dn}^{2}}\left(z,k\right)-{k^{\prime}}^{2}\right).$
22.13.4 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cd}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{cd}^{2}}\left(z,k\right)\right)\left(1-k^{2}{% \operatorname{cd}^{2}}\left(z,k\right)\right),$
22.13.5 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sd}\left(z,k\right)\right)^{% 2}={\left(1-{k^{\prime}}^{2}{\operatorname{sd}^{2}}\left(z,k\right)\right)}{% \left(1+k^{2}{\operatorname{sd}^{2}}\left(z,k\right)\right)},$
##### 9: 20.7 Identities
20.7.1 ${\theta_{3}^{2}}\left(0,q\right){\theta_{3}^{2}}\left(z,q\right)={\theta_{4}^{% 2}}\left(0,q\right){\theta_{4}^{2}}\left(z,q\right)+{\theta_{2}^{2}}\left(0,q% \right){\theta_{2}^{2}}\left(z,q\right),$
20.7.6 ${\theta_{4}^{2}}\left(0,q\right)\theta_{1}\left(w+z,q\right)\theta_{1}\left(w-% z,q\right)={\theta_{3}^{2}}\left(w,q\right){\theta_{2}^{2}}\left(z,q\right)-{% \theta_{2}^{2}}\left(w,q\right){\theta_{3}^{2}}\left(z,q\right),$
20.7.7 ${\theta_{4}^{2}}\left(0,q\right)\theta_{2}\left(w+z,q\right)\theta_{2}\left(w-% z,q\right)={\theta_{4}^{2}}\left(w,q\right){\theta_{2}^{2}}\left(z,q\right)-{% \theta_{1}^{2}}\left(w,q\right){\theta_{3}^{2}}\left(z,q\right),$
20.7.8 ${\theta_{4}^{2}}\left(0,q\right)\theta_{3}\left(w+z,q\right)\theta_{3}\left(w-% z,q\right)={\theta_{4}^{2}}\left(w,q\right){\theta_{3}^{2}}\left(z,q\right)-{% \theta_{1}^{2}}\left(w,q\right){\theta_{2}^{2}}\left(z,q\right),$
20.7.10 $\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z% ,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left% (0,q\right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}.$