# elliptic functions

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##### 2: 22.15 Inverse Functions
###### §22.15(i) Definitions
are denoted respectively by …The principal values satisfy …
##### 3: 22.2 Definitions
###### §22.2 Definitions
22.2.4 $\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{ns}\left(z,k\right)},$
22.2.9 $\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.$
As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … …
##### 5: 19.16 Definitions
All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …The $R$-function is often used to make a unified statement of a property of several elliptic integrals. …
###### §19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$
19.16.21 $R_{D}\left(0,y,z\right)=\tfrac{3}{4}\pi R_{-\frac{3}{2}}\left(\tfrac{1}{2},% \tfrac{3}{2};y,z\right),$
##### 6: 19.2 Definitions
19.2.4 $F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin% }^{2}\theta}}=\int_{0}^{\sin\phi}\frac{\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2% }t^{2}}},$
19.2.5 $E\left(\phi,k\right)=\int_{0}^{\phi}\sqrt{1-k^{2}{\sin}^{2}\theta}\mathrm{d}% \theta\\ =\int_{0}^{\sin\phi}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\mathrm{d}t.$
19.2.6 $D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin}^{2}\theta\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\mathrm{d}t}{% \sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))% /k^{2}.$
19.2.11_5 $\mathrm{el1}\left(x,k_{c}\right)=\int_{0}^{\operatorname{arctan}x}\frac{1}{% \sqrt{{\cos}^{2}\theta+k_{c}^{2}{\sin}^{2}\theta}}\mathrm{d}\theta,$
##### 7: 22.17 Moduli Outside the Interval [0,1]
###### §22.17(ii) Complex Moduli
When $z$ is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of $k^{2}$. …For proofs of these results and further information see Walker (2003).
See §22.17.