# §22.2 Definitions

The nome $q$ is given in terms of the modulus $k$ by

 22.2.1 $q=\mathop{\exp\/}\nolimits\!\left(-\pi\mathop{{K^{\prime}}\/}\nolimits\!\left(% k\right)/\mathop{K\/}\nolimits\!\left(k\right)\right),$ Defines: $q$: nome Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{{K^{\prime}}\/}\nolimits\!\left(\NVar{k}\right)$: Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function and $k$: modulus Referenced by: §20.9(i), §22.16(i) Permalink: http://dlmf.nist.gov/22.2.E1 Encodings: TeX, pMML, png See also: Annotations for 22.2

where $\mathop{K\/}\nolimits\!\left(k\right)$, $\mathop{{K^{\prime}}\/}\nolimits\!\left(k\right)$ are defined in §19.2(ii). Inversely,

 22.2.2 $\displaystyle k$ $\displaystyle=\frac{{\mathop{\theta_{2}\/}\nolimits^{2}}\!\left(0,q\right)}{{% \mathop{\theta_{3}\/}\nolimits^{2}}\!\left(0,q\right)},$ $\displaystyle k^{\prime}$ $\displaystyle=\frac{{\mathop{\theta_{4}\/}\nolimits^{2}}\!\left(0,q\right)}{{% \mathop{\theta_{3}\/}\nolimits^{2}}\!\left(0,q\right)},$ $\displaystyle\mathop{K\/}\nolimits\!\left(k\right)$ $\displaystyle=\frac{\pi}{2}{\mathop{\theta_{3}\/}\nolimits^{2}}\!\left(0,q% \right),$

where $k^{\prime}=\sqrt{1-k^{2}}$ and the theta functions are defined in §20.2(i).

With

 22.2.3 $\zeta=\frac{\pi z}{2\!\mathop{K\/}\nolimits\!\left(k\right)},$
 22.2.4 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=\frac{\mathop{\theta_{3}\/}% \nolimits\!\left(0,q\right)}{\mathop{\theta_{2}\/}\nolimits\!\left(0,q\right)}% \frac{\mathop{\theta_{1}\/}\nolimits\!\left(\zeta,q\right)}{\mathop{\theta_{4}% \/}\nolimits\!\left(\zeta,q\right)}=\frac{1}{\mathop{\mathrm{ns}\/}\nolimits% \left(z,k\right)},$ Defines: $\mathop{\mathrm{ns}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\mathop{\mathrm{sn}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/22.2.E4 Encodings: TeX, pMML, png See also: Annotations for 22.2
 22.2.5 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=\frac{\mathop{\theta_{4}\/}% \nolimits\!\left(0,q\right)}{\mathop{\theta_{2}\/}\nolimits\!\left(0,q\right)}% \frac{\mathop{\theta_{2}\/}\nolimits\!\left(\zeta,q\right)}{\mathop{\theta_{4}% \/}\nolimits\!\left(\zeta,q\right)}=\frac{1}{\mathop{\mathrm{nc}\/}\nolimits% \left(z,k\right)},$ Defines: $\mathop{\mathrm{cn}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\mathop{\mathrm{nc}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable Permalink: http://dlmf.nist.gov/22.2.E5 Encodings: TeX, pMML, png See also: Annotations for 22.2
 22.2.6 $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=\frac{\mathop{\theta_{4}\/}% \nolimits\!\left(0,q\right)}{\mathop{\theta_{3}\/}\nolimits\!\left(0,q\right)}% \frac{\mathop{\theta_{3}\/}\nolimits\!\left(\zeta,q\right)}{\mathop{\theta_{4}% \/}\nolimits\!\left(\zeta,q\right)}=\frac{1}{\mathop{\mathrm{nd}\/}\nolimits% \left(z,k\right)},$ Defines: $\mathop{\mathrm{dn}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\mathop{\mathrm{nd}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable Referenced by: §22.20(ii) Permalink: http://dlmf.nist.gov/22.2.E6 Encodings: TeX, pMML, png See also: Annotations for 22.2
 22.2.7 $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)=\frac{{\mathop{\theta_{3}\/}% \nolimits^{2}}\!\left(0,q\right)}{\mathop{\theta_{2}\/}\nolimits\!\left(0,q% \right)\mathop{\theta_{4}\/}\nolimits\!\left(0,q\right)}\frac{\mathop{\theta_{% 1}\/}\nolimits\!\left(\zeta,q\right)}{\mathop{\theta_{3}\/}\nolimits\!\left(% \zeta,q\right)}=\frac{1}{\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)},$ Defines: $\mathop{\mathrm{ds}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\mathop{\mathrm{sd}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable Permalink: http://dlmf.nist.gov/22.2.E7 Encodings: TeX, pMML, png See also: Annotations for 22.2
 22.2.8 $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)=\frac{\mathop{\theta_{3}\/}% \nolimits\!\left(0,q\right)}{\mathop{\theta_{2}\/}\nolimits\!\left(0,q\right)}% \frac{\mathop{\theta_{2}\/}\nolimits\!\left(\zeta,q\right)}{\mathop{\theta_{3}% \/}\nolimits\!\left(\zeta,q\right)}=\frac{1}{\mathop{\mathrm{dc}\/}\nolimits% \left(z,k\right)},$ Defines: $\mathop{\mathrm{cd}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\mathop{\mathrm{dc}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable Permalink: http://dlmf.nist.gov/22.2.E8 Encodings: TeX, pMML, png See also: Annotations for 22.2
 22.2.9 $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)=\frac{\mathop{\theta_{3}\/}% \nolimits\!\left(0,q\right)}{\mathop{\theta_{4}\/}\nolimits\!\left(0,q\right)}% \frac{\mathop{\theta_{1}\/}\nolimits\!\left(\zeta,q\right)}{\mathop{\theta_{2}% \/}\nolimits\!\left(\zeta,q\right)}=\frac{1}{\mathop{\mathrm{cs}\/}\nolimits% \left(z,k\right)}.$ Defines: $\mathop{\mathrm{cs}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\mathop{\mathrm{sc}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/22.2.E9 Encodings: TeX, pMML, png See also: Annotations for 22.2

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. Each is meromorphic in $z$ for fixed $k$, with simple poles and simple zeros, and each is meromorphic in $k$ for fixed $z$. For $k\in[0,1]$, all functions are real for $z\in\mathbb{R}$.

## Glaisher’s Notation

The Jacobian functions are related in the following way. Let $\mathrm{p}$, $\mathrm{q}$, $\mathrm{r}$ be any three of the letters $\mathrm{s}$, $\mathrm{c}$, $\mathrm{d}$, $\mathrm{n}$. Then

 22.2.10 $\mathop{\mathrm{p\!q}\/}\nolimits\left(z,k\right)=\frac{\mathop{\mathrm{p\!r}% \/}\nolimits\left(z,k\right)}{\mathop{\mathrm{q\!r}\/}\nolimits\left(z,k\right% )}=\frac{1}{\mathop{\mathrm{q\!p}\/}\nolimits\left(z,k\right)},$ Defines: $\mathop{\mathrm{p\!q}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: generic Jacobian elliptic function Symbols: $z$: complex and $k$: modulus A&S Ref: 16.3.4 Referenced by: §22.14(iv), §22.20(ii), §22.4(ii), §22.5(i) Permalink: http://dlmf.nist.gov/22.2.E10 Encodings: TeX, pMML, png See also: Annotations for 22.2

with the convention that functions with the same two letters are replaced by unity; e.g. $\mathop{\mathrm{s\!s}\/}\nolimits\left(z,k\right)=1$.

The six functions containing the letter $\mathrm{s}$ in their two-letter name are odd in $z$; the other six are even in $z$.

In terms of Neville’s theta functions (§20.1)

 22.2.11 $\mathop{\mathrm{p\!q}\/}\nolimits\left(z,k\right)=\ifrac{\mathop{\theta_{p}\/}% \nolimits\!\left(z\middle|\tau\right)}{\mathop{\theta_{q}\/}\nolimits\!\left(z% \middle|\tau\right)},$

where

 22.2.12 $\tau=\ifrac{\mathrm{i}\mathop{{K^{\prime}}\/}\nolimits\!\left(k\right)}{% \mathop{K\/}\nolimits\!\left(k\right)},$

and on the left-hand side of (22.2.11) $\mathrm{p}$, $\mathrm{q}$ are any pair of the letters $\mathrm{s}$, $\mathrm{c}$, $\mathrm{d}$, $\mathrm{n}$, and on the right-hand side they correspond to the integers $1,2,3,4$.