elliptic functions

(0.007 seconds)

1—10 of 164 matching pages

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 $F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{\sqrt{1-k^{2}{% \sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{% 1-k^{2}t^{2}}},$
19.2.6 $D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin}^{2}\theta\,\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\,\mathrm{d}t}{% \sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))% /k^{2}.$
The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. …
19.2.11_5 $\operatorname{el1}\left(x,k_{c}\right)=\int_{0}^{\operatorname{arctan}x}\frac{% 1}{\sqrt{{\cos}^{2}\theta+k_{c}^{2}{\sin}^{2}\theta}}\,\mathrm{d}\theta,$
19.2.16 $\operatorname{el3}\left(x,k_{c},p\right)=\int_{0}^{\operatorname{arctan}x}% \frac{\,\mathrm{d}\theta}{({\cos}^{2}\theta+p{\sin}^{2}\theta)\sqrt{{\cos}^{2}% \theta+k_{c}^{2}{\sin}^{2}\theta}}=\Pi\left(\operatorname{arctan}x,1-p,k\right),$ $x^{2}\neq-1/p$.
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.