# §22.2 Definitions

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

 22.2.1 $q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right),$ ⓘ Defines: $q$: nome Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $\exp\NVar{z}$: 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 and Ch.22

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

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

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

With

 22.2.3 $\zeta=\frac{\pi z}{2K\left(k\right)},$
 22.2.4 $\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{ns}\left(z,k\right)},$ ⓘ Defines: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: 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 and Ch.22
 22.2.5 $\operatorname{cn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nc}\left(z,k\right)},$ ⓘ Defines: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: 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 and Ch.22
 22.2.6 $\operatorname{dn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{3}% \left(0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nd}\left(z,k\right)},$ ⓘ Defines: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: 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 and Ch.22
 22.2.7 $\operatorname{sd}\left(z,k\right)=\frac{{\theta_{3}^{2}}\left(0,q\right)}{% \theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(% \zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{ds}% \left(z,k\right)},$ ⓘ Defines: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: 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 and Ch.22
 22.2.8 $\operatorname{cd}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q% \right)}=\frac{1}{\operatorname{dc}\left(z,k\right)},$ ⓘ Defines: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: 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 and Ch.22
 22.2.9 $\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.$ ⓘ Defines: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function and $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: 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 and Ch.22

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 $\operatorname{pq}\left(z,k\right)=\frac{\operatorname{pr}\left(z,k\right)}{% \operatorname{qr}\left(z,k\right)}=\frac{1}{\operatorname{qp}\left(z,k\right)},$ ⓘ Defines: $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$: 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, §22.2 and Ch.22

with the convention that functions with the same two letters are replaced by unity; e.g. $\operatorname{ss}\left(z,k\right)=1$.

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 $\operatorname{pq}\left(z,k\right)=\ifrac{\theta_{p}\left(z\middle|\tau\right)}% {\theta_{q}\left(z\middle|\tau\right)},$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$: generic Jacobian elliptic function, $q$: nome, $z$: complex, $k$: modulus and $\tau$: ratio A&S Ref: 16.36.3 Referenced by: §22.2 Permalink: http://dlmf.nist.gov/22.2.E11 Encodings: TeX, pMML, png See also: Annotations for §22.2, §22.2 and Ch.22

where

 22.2.12 $\tau=\ifrac{\mathrm{i}{K^{\prime}}\left(k\right)}{K\left(k\right)},$

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 $1,2,3,4$.