# Jacobi epsilon function

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##### 2: 22.1 Special Notation
The functions treated in this chapter are the three principal Jacobian elliptic functions $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$; the nine subsidiary Jacobian elliptic functions $\operatorname{cd}\left(z,k\right)$, $\operatorname{sd}\left(z,k\right)$, $\operatorname{nd}\left(z,k\right)$, $\operatorname{dc}\left(z,k\right)$, $\operatorname{nc}\left(z,k\right)$, $\operatorname{sc}\left(z,k\right)$, $\operatorname{ns}\left(z,k\right)$, $\operatorname{ds}\left(z,k\right)$, $\operatorname{cs}\left(z,k\right)$; the amplitude function $\operatorname{am}\left(x,k\right)$; Jacobi’s epsilon and zeta functions $\mathcal{E}\left(x,k\right)$ and $\mathrm{Z}\left(x|k\right)$. …
##### 4: 22.18 Mathematical Applications
###### Ellipse
where $\mathcal{E}\left(u,k\right)$ is Jacobi’s epsilon function22.16(ii)). …
##### 5: 22.20 Methods of Computation
Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. … …
##### 6: 31.16 Mathematical Applications
31.16.1 $\mathit{Hp}_{n,m}\left(x\right)\mathit{Hp}_{n,m}\left(y\right)=\sum_{j=0}^{n}A% _{j}{\sin}^{2j}\theta\*P^{(\gamma+\delta+2j-1,\epsilon-1)}_{n-j}\left(\cos% \left(2\theta\right)\right)P^{(\delta-1,\gamma-1)}_{j}\left(\cos\left(2\phi% \right)\right),$
##### 7: 31.7 Relations to Other Functions
###### §31.7(ii) Relations to Lamé Functions
With $z={\operatorname{sn}}^{2}\left(\zeta,k\right)$ and …
$\gamma=\delta=\epsilon=\tfrac{1}{2},$
##### 8: 31.2 Differential Equations
31.2.8 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+\left((2\gamma-1)\frac{% \operatorname{cn}\zeta\operatorname{dn}\zeta}{\operatorname{sn}\zeta}-(2\delta% -1)\frac{\operatorname{sn}\zeta\operatorname{dn}\zeta}{\operatorname{cn}\zeta}% -(2\epsilon-1)k^{2}\frac{\operatorname{sn}\zeta\operatorname{cn}\zeta}{% \operatorname{dn}\zeta}\right)\frac{\mathrm{d}w}{\mathrm{d}\zeta}+4k^{2}(% \alpha\beta{\operatorname{sn}}^{2}\zeta-q)w=0.$
##### 9: 18.15 Asymptotic Approximations
18.15.6 $(\sin\tfrac{1}{2}\theta)^{\alpha+\frac{1}{2}}(\cos\tfrac{1}{2}\theta)^{\beta+% \frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)=\frac{\Gamma\left(n+% \alpha+1\right)}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*\left(\theta^{\frac{1}{2}}J_% {\alpha}\left(\rho\theta\right)\sum_{m=0}^{M}\dfrac{A_{m}(\theta)}{\rho^{2m}}+% \theta^{\frac{3}{2}}J_{\alpha+1}\left(\rho\theta\right)\sum_{m=0}^{M-1}\dfrac{% B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M}(\rho,\theta)\right),$
##### 10: 20.15 Tables
Tables of Neville’s theta functions $\theta_{s}\left(x,q\right)$, $\theta_{c}\left(x,q\right)$, $\theta_{d}\left(x,q\right)$, $\theta_{n}\left(x,q\right)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\varepsilon,\alpha=0(5^{\circ})90^{\circ}$, where (in radian measure) $\varepsilon=x/{\theta_{3}}^{2}\left(0,q\right)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …