§22.2 Definitions

Notes:: See Lawden (1989, §2.1), Walker (1996, §§5.1, 6.2), Walker (2003), Whittaker and Watson (1927, §22.1).
Keywords:: Jacobian elliptic functions, Neville’s theta functions, nome, theta functions
Referenced by:: §19.25(v), ¶ ‣ §20.1, §20.11(iii), §20.7(ii), §20.9(ii), §22.11, §22.20(i), §22.21, §29.2(i), §29.3(v), §29.6(i), §29.8, ¶ ‣ §31.2(iv)
Permalink:: http://dlmf.nist.gov/22.2

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:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\exp\/}\nolimits z$ : exponential function and $k$ : modulus
Referenced by:: §20.9(i), ¶ ‣ §22.16(i)
Permalink:: http://dlmf.nist.gov/22.2.E1
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where $\mathop{K\/}\nolimits\!\left(k\right)$ , $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ are defined in §19.2(ii). Inversely,

22.2.2

$k=\frac{{\mathop{\theta _{{2}}\/}\nolimits^{{2}}}\!\left(0,q\right)}{{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right)},$

$k^{{\prime}}=\frac{{\mathop{\theta _{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right)}{{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right)},$

$\mathop{K\/}\nolimits\!\left(k\right)=\frac{\pi}{2}{\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $q$ : nome, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/22.2.E2
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where $k^{{\prime}}=\sqrt{1-k^{2}}$ and the theta functions are defined in §20.2(i).

With

22.2.3 $\zeta=\frac{\pi z}{2\!\mathop{K\/}\nolimits\!\left(k\right)},$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E3
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function and $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function
Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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
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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:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function and $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function
Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

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:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function and $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function
Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E6
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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:: $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function and $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function
Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

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:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function and $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function
Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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

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:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function and $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function
Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : 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
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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 $z\in\Real$ .

¶ Glaisher’s Notation

Keywords:: theta functions

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 $\mathop{\mathrm{pq}\/}\nolimits\left(z,k\right)=\frac{\mathop{\mathrm{pr}\/}\nolimits\left(z,k\right)}{\mathop{\mathrm{qr}\/}\nolimits\left(z,k\right)}=\frac{1}{\mathop{\mathrm{qp}\/}\nolimits\left(z,k\right)},$

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

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 $\mathop{\mathrm{pq}\/}\nolimits\left(z,k\right)=\ifrac{\mathop{\theta _{{p}}\/}\nolimits\!\left(z\middle|\tau\right)}{\mathop{\theta _{{q}}\/}\nolimits\!\left(z\middle|\tau\right)},$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\mathrm{pq}\/}\nolimits\left(z,k\right)$ : generic Jacobian elliptic function, $q$ : nome, $z$ : complex, $k$ : modulus and $\tau$ : ratio
A&S Ref:: 16.36.3
Permalink:: http://dlmf.nist.gov/22.2.E11
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where

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

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $k$ : modulus and $\tau$ : ratio
Permalink:: http://dlmf.nist.gov/22.2.E12
Encodings:: TeX, pMML, png

and $\mathrm{p}$ , $\mathrm{q}$ are any pair of the letters $\mathrm{s}$ , $\mathrm{c}$ , $\mathrm{d}$ , $\mathrm{n}$ .