# 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 $\mathrm{Z}\left(x|k\right)$. …
##### 3: 33.14 Definitions and Basic Properties
###### §33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$
For nonzero values of $\epsilon$ and $r$ the function $h\left(\epsilon,\ell;r\right)$ is defined by …
###### §33.14(iv) Solutions $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$
The function $s\left(\epsilon,\ell;r\right)$ has the following properties: …
##### 5: 22.18 Mathematical Applications
###### Ellipse
where $\mathcal{E}\left(u,k\right)$ is Jacobi’s epsilon function22.16(ii)). …
##### 8: 33.20 Expansions for Small $|\epsilon|$
###### §33.20(iii) Asymptotic Expansion for the Irregular Solution
where $A(\epsilon,\ell)$ is given by (33.14.11), (33.14.12), and …
##### 9: 33.17 Recurrence Relations and Derivatives
###### §33.17 Recurrence Relations and Derivatives
33.17.1 $(\ell+1)rf\left(\epsilon,\ell-1;r\right)-(2\ell+1)\left(\ell(\ell+1)-r\right)f% \left(\epsilon,\ell;r\right)+\ell\left(1+(\ell+1)^{2}\epsilon\right)rf\left(% \epsilon,\ell+1;r\right)=0,$
33.17.2 $(\ell+1)\left(1+\ell^{2}\epsilon\right)rh\left(\epsilon,\ell-1;r\right)-(2\ell% +1)\left(\ell(\ell+1)-r\right)h\left(\epsilon,\ell;r\right)+\ell rh\left(% \epsilon,\ell+1;r\right)=0,$
33.17.3 $(\ell+1)rf'\left(\epsilon,\ell;r\right)=\left((\ell+1)^{2}-r\right)f\left(% \epsilon,\ell;r\right)-\left(1+(\ell+1)^{2}\epsilon\right)rf\left(\epsilon,% \ell+1;r\right),$
33.17.4 $(\ell+1)rh'\left(\epsilon,\ell;r\right)=\left((\ell+1)^{2}-r\right)h\left(% \epsilon,\ell;r\right)-rh\left(\epsilon,\ell+1;r\right).$
##### 10: 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. … …