§22.8 Addition Theorems

Keywords:: Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.8

§22.8(i) Sum of Two Arguments
§22.8(ii) Alternative Forms for Sum of Two Arguments
§22.8(iii) Special Relations Between Arguments

§22.8(i) Sum of Two Arguments

Notes:: See Lawden (1989, pp. 33–36), Walker (1996, pp. 126–131), Whittaker and Watson (1927, pp. 494–498).
Permalink:: http://dlmf.nist.gov/22.8.i

For $u,v\in\Complex$ , and with the common modulus $k$ suppressed:

22.8.1 $\mathop{\mathrm{sn}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits v+\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits u}{1-k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}v},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
A&S Ref:: 16.17.1
Referenced by:: §22.18(iv)
Permalink:: http://dlmf.nist.gov/22.8.E1
Encodings:: TeX, pMML, png

22.8.2 $\mathop{\mathrm{cn}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v-\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits u\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits v}{1-k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}v},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
A&S Ref:: 16.17.2
Referenced by:: §22.6(ii)
Permalink:: http://dlmf.nist.gov/22.8.E2
Encodings:: TeX, pMML, png

22.8.3 $\mathop{\mathrm{dn}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{dn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits v-k^{2}\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{cn}\/}\nolimits v}{1-k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}v}.$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
A&S Ref:: 16.17.3
Permalink:: http://dlmf.nist.gov/22.8.E3
Encodings:: TeX, pMML, png

22.8.4 $\mathop{\mathrm{cd}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{cd}\/}\nolimits u\mathop{\mathrm{cd}\/}\nolimits v-{k^{{\prime}}}^{2}\mathop{\mathrm{sd}\/}\nolimits u\mathop{\mathrm{nd}\/}\nolimits u\mathop{\mathrm{sd}\/}\nolimits v\mathop{\mathrm{nd}\/}\nolimits v}{1+k^{2}{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}v},$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E4
Encodings:: TeX, pMML, png

22.8.5 $\mathop{\mathrm{sd}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{sd}\/}\nolimits u\mathop{\mathrm{cd}\/}\nolimits v\mathop{\mathrm{nd}\/}\nolimits v+\mathop{\mathrm{sd}\/}\nolimits v\mathop{\mathrm{cd}\/}\nolimits u\mathop{\mathrm{nd}\/}\nolimits u}{1+k^{2}{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}v},$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E5
Encodings:: TeX, pMML, png

22.8.6 $\mathop{\mathrm{nd}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{nd}\/}\nolimits u\mathop{\mathrm{nd}\/}\nolimits v+k^{2}\mathop{\mathrm{sd}\/}\nolimits u\mathop{\mathrm{cd}\/}\nolimits u\mathop{\mathrm{sd}\/}\nolimits v\mathop{\mathrm{cd}\/}\nolimits v}{1+k^{2}{k^{{\prime}}}^{2}{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sd}\/}\nolimits^{{2}}}v},$

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E6
Encodings:: TeX, pMML, png

22.8.7 $\mathop{\mathrm{dc}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{dc}\/}\nolimits u\mathop{\mathrm{dc}\/}\nolimits v+{k^{{\prime}}}^{2}\mathop{\mathrm{sc}\/}\nolimits u\mathop{\mathrm{nc}\/}\nolimits u\mathop{\mathrm{sc}\/}\nolimits v\mathop{\mathrm{nc}\/}\nolimits v}{1-{k^{{\prime}}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}v},$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E7
Encodings:: TeX, pMML, png

22.8.8 $\mathop{\mathrm{nc}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{nc}\/}\nolimits u\mathop{\mathrm{nc}\/}\nolimits v+\mathop{\mathrm{sc}\/}\nolimits u\mathop{\mathrm{dc}\/}\nolimits u\mathop{\mathrm{sc}\/}\nolimits v\mathop{\mathrm{dc}\/}\nolimits v}{1-{k^{{\prime}}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}v},$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E8
Encodings:: TeX, pMML, png

22.8.9 $\mathop{\mathrm{sc}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{sc}\/}\nolimits u\mathop{\mathrm{dc}\/}\nolimits v\mathop{\mathrm{nc}\/}\nolimits v+\mathop{\mathrm{sc}\/}\nolimits v\mathop{\mathrm{dc}\/}\nolimits u\mathop{\mathrm{nc}\/}\nolimits u}{1-{k^{{\prime}}}^{2}{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sc}\/}\nolimits^{{2}}}v},$

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E9
Encodings:: TeX, pMML, png

22.8.10 $\mathop{\mathrm{ns}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{ns}\/}\nolimits u\mathop{\mathrm{ds}\/}\nolimits v\mathop{\mathrm{cs}\/}\nolimits v-\mathop{\mathrm{ns}\/}\nolimits v\mathop{\mathrm{ds}\/}\nolimits u\mathop{\mathrm{cs}\/}\nolimits u}{{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}v-{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}u},$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E10
Encodings:: TeX, pMML, png

22.8.11 $\mathop{\mathrm{ds}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{ds}\/}\nolimits u\mathop{\mathrm{cs}\/}\nolimits v\mathop{\mathrm{ns}\/}\nolimits v-\mathop{\mathrm{ds}\/}\nolimits v\mathop{\mathrm{cs}\/}\nolimits u\mathop{\mathrm{ns}\/}\nolimits u}{{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}v-{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}u},$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E11
Encodings:: TeX, pMML, png

22.8.12 $\mathop{\mathrm{cs}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{cs}\/}\nolimits u\mathop{\mathrm{ds}\/}\nolimits v\mathop{\mathrm{ns}\/}\nolimits v-\mathop{\mathrm{cs}\/}\nolimits v\mathop{\mathrm{ds}\/}\nolimits u\mathop{\mathrm{ns}\/}\nolimits u}{{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}v-{\mathop{\mathrm{cs}\/}\nolimits^{{2}}}u}.$

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E12
Encodings:: TeX, pMML, png

§22.8(ii) Alternative Forms for Sum of Two Arguments

Notes:: See Lawden (1989, pp. 33–36, 43), Whittaker and Watson (1927, pp. 494–498).
Permalink:: http://dlmf.nist.gov/22.8.ii

For $u,v\in\Complex$ , and with the common modulus $k$ suppressed:

22.8.13 $\mathop{\mathrm{sn}\/}\nolimits(u+v)=\frac{{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}u-{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}v}{\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits v-\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits u},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E13
Encodings:: TeX, pMML, png

22.8.14 $\mathop{\mathrm{sn}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits v+\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{cn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits u}{\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v+\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits u\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits v},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E14
Encodings:: TeX, pMML, png

22.8.15 $\mathop{\mathrm{cn}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits v-\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{cn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits u}{\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits v-\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits u},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E15
Encodings:: TeX, pMML, png

22.8.16 $\mathop{\mathrm{cn}\/}\nolimits(u+v)=\frac{1-{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}u-{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}v+k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}u{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}v}{\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v+\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits u\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits v},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E16
Encodings:: TeX, pMML, png

22.8.17 $\mathop{\mathrm{dn}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits u-\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits v}{\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits v-\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits u},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E17
Encodings:: TeX, pMML, png

22.8.18 $\mathop{\mathrm{dn}\/}\nolimits(u+v)=\frac{\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits v+{k^{{\prime}}}^{2}\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{sn}\/}\nolimits v}{\mathop{\mathrm{cn}\/}\nolimits u\mathop{\mathrm{cn}\/}\nolimits v+\mathop{\mathrm{sn}\/}\nolimits u\mathop{\mathrm{dn}\/}\nolimits u\mathop{\mathrm{sn}\/}\nolimits v\mathop{\mathrm{dn}\/}\nolimits v}.$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E18
Encodings:: TeX, pMML, png

§22.8(iii) Special Relations Between Arguments

Notes:: See Whittaker and Watson (1927, p. 530).
Keywords:: Jacobian elliptic functions, poristic polygon constructions of Poncelet
Permalink:: http://dlmf.nist.gov/22.8.iii

In the following equations the common modulus $k$ is again suppressed.

Let

22.8.19 $z_{1}+z_{2}+z_{3}+z_{4}=0.$

Symbols:: $z$ : complex
Permalink:: http://dlmf.nist.gov/22.8.E19
Encodings:: TeX, pMML, png

Then

22.8.20 $\begin{vmatrix}\mathop{\mathrm{sn}\/}\nolimits z_{1}&\mathop{\mathrm{cn}\/}\nolimits z_{1}&\mathop{\mathrm{dn}\/}\nolimits z_{1}&1\\ \mathop{\mathrm{sn}\/}\nolimits z_{2}&\mathop{\mathrm{cn}\/}\nolimits z_{2}&\mathop{\mathrm{dn}\/}\nolimits z_{2}&1\\ \mathop{\mathrm{sn}\/}\nolimits z_{3}&\mathop{\mathrm{cn}\/}\nolimits z_{3}&\mathop{\mathrm{dn}\/}\nolimits z_{3}&1\\ \mathop{\mathrm{sn}\/}\nolimits z_{4}&\mathop{\mathrm{cn}\/}\nolimits z_{4}&\mathop{\mathrm{dn}\/}\nolimits z_{4}&1\end{vmatrix}=0,$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
Referenced by:: §22.8(iii)
Permalink:: http://dlmf.nist.gov/22.8.E20
Encodings:: TeX, pMML, png

and

22.8.21 ${k^{{\prime}}}^{2}-{k^{{\prime}}}^{2}k^{2}\mathop{\mathrm{sn}\/}\nolimits z_{1}\mathop{\mathrm{sn}\/}\nolimits z_{2}\mathop{\mathrm{sn}\/}\nolimits z_{3}\mathop{\mathrm{sn}\/}\nolimits z_{4}+k^{2}\mathop{\mathrm{cn}\/}\nolimits z_{1}\mathop{\mathrm{cn}\/}\nolimits z_{2}\mathop{\mathrm{cn}\/}\nolimits z_{3}\mathop{\mathrm{cn}\/}\nolimits z_{4}-\mathop{\mathrm{dn}\/}\nolimits z_{1}\mathop{\mathrm{dn}\/}\nolimits z_{2}\mathop{\mathrm{dn}\/}\nolimits z_{3}\mathop{\mathrm{dn}\/}\nolimits z_{4}=0.$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.8.E21
Encodings:: TeX, pMML, png

A geometric interpretation of (22.8.20) analogous to that of (23.10.5) is given in Whittaker and Watson (1927, p. 530).

Next, let

22.8.22 $z_{1}+z_{2}+z_{3}+z_{4}=2\!\mathop{K\/}\nolimits\!\left(k\right).$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E22
Encodings:: TeX, pMML, png

Then

22.8.23 $\begin{vmatrix}\mathop{\mathrm{sn}\/}\nolimits z_{1}\mathop{\mathrm{cn}\/}\nolimits z_{1}&\mathop{\mathrm{cn}\/}\nolimits z_{1}\mathop{\mathrm{dn}\/}\nolimits z_{1}&\mathop{\mathrm{cn}\/}\nolimits z_{1}&\mathop{\mathrm{dn}\/}\nolimits z_{1}\\ \mathop{\mathrm{sn}\/}\nolimits z_{2}\mathop{\mathrm{cn}\/}\nolimits z_{2}&\mathop{\mathrm{cn}\/}\nolimits z_{2}\mathop{\mathrm{dn}\/}\nolimits z_{2}&\mathop{\mathrm{cn}\/}\nolimits z_{2}&\mathop{\mathrm{dn}\/}\nolimits z_{2}\\ \mathop{\mathrm{sn}\/}\nolimits z_{3}\mathop{\mathrm{cn}\/}\nolimits z_{3}&\mathop{\mathrm{cn}\/}\nolimits z_{3}\mathop{\mathrm{dn}\/}\nolimits z_{3}&\mathop{\mathrm{cn}\/}\nolimits z_{3}&\mathop{\mathrm{dn}\/}\nolimits z_{3}\\ \mathop{\mathrm{sn}\/}\nolimits z_{4}\mathop{\mathrm{cn}\/}\nolimits z_{4}&\mathop{\mathrm{cn}\/}\nolimits z_{4}\mathop{\mathrm{dn}\/}\nolimits z_{4}&\mathop{\mathrm{cn}\/}\nolimits z_{4}&\mathop{\mathrm{dn}\/}\nolimits z_{4}\end{vmatrix}=0.$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E23
Encodings:: TeX, pMML, png

For these and related identities see Copson (1935, pp. 415–416).

If sums/differences of the $z_{j}$ ’s are rational multiples of $\mathop{K\/}\nolimits\!\left(k\right)$ , then further relations follow. For instance, if

22.8.24 $z_{1}-z_{2}=z_{2}-z_{3}=\tfrac{2}{3}\!\mathop{K\/}\nolimits\!\left(k\right),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E24
Encodings:: TeX, pMML, png

then

22.8.25 $\frac{(\mathop{\mathrm{dn}\/}\nolimits z_{2}+\mathop{\mathrm{dn}\/}\nolimits z_{3})(\mathop{\mathrm{dn}\/}\nolimits z_{3}+\mathop{\mathrm{dn}\/}\nolimits z_{1})(\mathop{\mathrm{dn}\/}\nolimits z_{1}+\mathop{\mathrm{dn}\/}\nolimits z_{2})}{\mathop{\mathrm{dn}\/}\nolimits z_{1}+\mathop{\mathrm{dn}\/}\nolimits z_{2}+\mathop{\mathrm{dn}\/}\nolimits z_{3}}$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E25
Encodings:: TeX, pMML, png

is independent of $z_{1}$ , $z_{2}$ , $z_{3}$ . Similarly, if

22.8.26 $z_{1}-z_{2}=z_{2}-z_{3}=z_{3}-z_{4}=\tfrac{1}{2}\!\mathop{K\/}\nolimits\!\left(k\right),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E26
Encodings:: TeX, pMML, png

then

22.8.27 $\mathop{\mathrm{dn}\/}\nolimits z_{1}\mathop{\mathrm{dn}\/}\nolimits z_{3}=\mathop{\mathrm{dn}\/}\nolimits z_{2}\mathop{\mathrm{dn}\/}\nolimits z_{4}=k^{{\prime}}.$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.8.E27
Encodings:: TeX, pMML, png

Greenhill (1959, pp. 121–130) reviews these results in terms of the geometric poristic polygon constructions of Poncelet. Generalizations are given in §22.9.

§22.8 Addition Theorems

Contents

§22.8(i) Sum of Two Arguments

§22.8(ii) Alternative Forms for Sum of Two Arguments

§22.8(iii) Special Relations Between Arguments