# Jacobi elliptic form

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##### 2: 22.5 Special Values
For example, at $z=K+iK^{\prime}$, $\operatorname{sn}\left(z,k\right)=1/k$, $\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0$. … Table 22.5.2 gives $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$ for other special values of $z$. For example, $\operatorname{sn}\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}$. …
##### 3: 22.18 Mathematical Applications
For any two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ on this curve, their sum $(x_{3},y_{3})$, always a third point on the curve, is defined by the Jacobi–Abel addition law …
##### 4: Errata
• Table 22.5.4

Originally the limiting form for $\operatorname{sc}\left(z,k\right)$ in the last line of this table was incorrect ($\cosh z$, instead of $\sinh z$).

Reported 2010-11-23.

• ##### 5: 29.12 Definitions
With the substitution $\xi={\operatorname{sn}}^{2}\left(z,k\right)$ every Lamé polynomial in Table 29.12.1 can be written in the form
##### 6: 29.15 Fourier Series and Chebyshev Series
Since (29.2.5) implies that $\cos\phi=\operatorname{sn}\left(z,k\right)$, (29.15.1) can be rewritten in the form
###### §22.8(iii) Special Relations Between Arguments
If sums/differences of the $z_{j}$’s are rational multiples of $K\left(k\right)$, then further relations follow. …
##### 8: 29.2 Differential Equations
For $\operatorname{sn}\left(z,k\right)$ see §22.2. …
###### §29.2(ii) Other Forms
For $\operatorname{am}\left(z,k\right)$ see §22.16(i). … we have …
##### 9: 22.15 Inverse Functions
###### §22.15 Inverse Functions
are denoted respectively by …
###### §22.15(ii) Representations as Elliptic Integrals
The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. …
##### 10: 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 $z=2mK+(2n+1)iK^{\prime}$. … Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form $2mK+2niK^{\prime}$, where $m,n\in\mathbb{Z}$. … 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 $\operatorname{pq}\left(z,k\right)$. … For example, $\operatorname{sn}\left(z+K,k\right)=\operatorname{cd}\left(z,k\right)$. …