# general elliptic functions

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##### 1: 19.2 Definitions
###### §19.2(ii) Legendre’s Integrals
The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. …
##### 2: 21.8 Abelian Functions
In consequence, Abelian functions are generalizations of elliptic functions23.2(iii)) to more than one complex variable. …
##### 3: 23.2 Definitions and Periodic Properties
23.2.9 $\wp\left(z+2\omega_{j}\right)=\wp\left(z\right),$ $j=1,2,3$.
##### 4: 23.6 Relations to Other Functions
23.6.21 $\wp\left(z\right)-e_{1}=\frac{{K}^{2}}{\omega_{1}^{2}}{\operatorname{cs}}^{2}% \left(\frac{K\!z}{\omega_{1}},k\right),$
23.6.22 $\wp\left(z\right)-e_{2}=\frac{{K}^{2}}{\omega_{1}^{2}}{\operatorname{ds}}^{2}% \left(\frac{K\!z}{\omega_{1}},k\right),$
23.6.23 $\wp\left(z\right)-e_{3}=\frac{{K}^{2}}{\omega_{1}^{2}}{\operatorname{ns}}^{2}% \left(\frac{K\!z}{\omega_{1}},k\right).$
###### §23.6(iii) GeneralEllipticFunctions
For representations of general elliptic functions23.2(iii)) in terms of $\sigma\left(z\right)$ and $\wp\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta\left(z\right)$ see Lawden (1989, §8.11). …
##### 5: 22.2 Definitions
22.2.10 $\operatorname{pq}\left(z,k\right)=\frac{\operatorname{pr}\left(z,k\right)}{% \operatorname{qr}\left(z,k\right)}=\frac{1}{\operatorname{qp}\left(z,k\right)},$
##### 6: 22.17 Moduli Outside the Interval [0,1]
22.17.1 $\operatorname{pq}\left(z,k\right)=\operatorname{pq}\left(z,-k\right),$
##### 8: 22.6 Elementary Identities
22.6.22 ${\operatorname{pq}}^{2}\left(\tfrac{1}{2}z,k\right)=\frac{\operatorname{ps}% \left(z,k\right)+\operatorname{rs}\left(z,k\right)}{\operatorname{qs}\left(z,k% \right)+\operatorname{rs}\left(z,k\right)}=\frac{\operatorname{pq}\left(z,k% \right)+\operatorname{rq}\left(z,k\right)}{1+\operatorname{rq}\left(z,k\right)% }=\frac{\operatorname{pr}\left(z,k\right)+1}{\operatorname{qr}\left(z,k\right)% +1}.$
##### 9: 19.25 Relations to Other Functions
19.25.28 $\Delta(\mathrm{p,q})={\operatorname{ps}}^{2}\left(u,k\right)-{\operatorname{qs% }}^{2}\left(u,k\right)=-\Delta(\mathrm{q,p}),$
19.25.31 $u=R_{F}\left({\operatorname{ps}}^{2}\left(u,k\right),{\operatorname{qs}}^{2}% \left(u,k\right),{\operatorname{rs}}^{2}\left(u,k\right)\right);$
##### 10: 22.14 Integrals
22.14.2 $\int\operatorname{cn}\left(x,k\right)\,\mathrm{d}x=k^{-1}\operatorname{Arccos}% \left(\operatorname{dn}\left(x,k\right)\right),$
22.14.3 $\int\operatorname{dn}\left(x,k\right)\,\mathrm{d}x=\operatorname{Arcsin}\left(% \operatorname{sn}\left(x,k\right)\right)=\operatorname{am}\left(x,k\right).$
22.14.5 $\int\operatorname{sd}\left(x,k\right)\,\mathrm{d}x=(kk^{\prime})^{-1}% \operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right)\right),$
22.14.6 $\int\operatorname{nd}\left(x,k\right)\,\mathrm{d}x={k^{\prime}}^{-1}% \operatorname{Arccos}\left(\operatorname{cd}\left(x,k\right)\right).$