22 Jacobian Elliptic Functions Properties 22.7 Landen Transformations 22.9 Cyclic Identities

§22.8 Addition Theorems

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Keywords:: Jacobian elliptic functions, addition theorems
Permalink:: http://dlmf.nist.gov/22.8
See also:: Annotations for Ch.22

§22.8(i) Sum of Two Arguments

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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
See also:: Annotations for §22.8 and Ch.22

For $u,v\in\mathbb{C}$ , and with the common modulus $k$ suppressed:

22.8.1	$\displaystyle\operatorname{sn}(u v)$	$\displaystyle=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}v % \operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}{1-k^{2}{\operatorname{% sn}}^{2}u{\operatorname{sn}}^{2}v},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : 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 See also: Annotations for §22.8(i), §22.8 and Ch.22
22.8.2	$\displaystyle\operatorname{cn}(u v)$	$\displaystyle=\frac{\operatorname{cn}u\operatorname{cn}v-\operatorname{sn}u% \operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}{1-k^{2}{\operatorname{% sn}}^{2}u{\operatorname{sn}}^{2}v},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : 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 See also: Annotations for §22.8(i), §22.8 and Ch.22
22.8.3	$\displaystyle\operatorname{dn}(u v)$	$\displaystyle=\frac{\operatorname{dn}u\operatorname{dn}v-k^{2}\operatorname{sn% }u\operatorname{cn}u\operatorname{sn}v\operatorname{cn}v}{1-k^{2}{% \operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v}.$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : 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 See also: Annotations for §22.8(i), §22.8 and Ch.22

22.8.4		$\operatorname{cd}(u v)=\frac{\operatorname{cd}u\operatorname{cd}v-{k^{\prime}}% ^{2}\operatorname{sd}u\operatorname{nd}u\operatorname{sd}v\operatorname{nd}v}{% 1 k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E4 Encodings: TeX, pMML, png See also: Annotations for §22.8(i), §22.8 and Ch.22

22.8.5	$\displaystyle\operatorname{sd}(u v)$	$\displaystyle=\frac{\operatorname{sd}u\operatorname{cd}v\operatorname{nd}v % \operatorname{sd}v\operatorname{cd}u\operatorname{nd}u}{1 k^{2}{k^{\prime}}^{2% }{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E5 Encodings: TeX, pMML, png See also: Annotations for §22.8(i), §22.8 and Ch.22
22.8.6	$\displaystyle\operatorname{nd}(u v)$	$\displaystyle=\frac{\operatorname{nd}u\operatorname{nd}v k^{2}\operatorname{sd% }u\operatorname{cd}u\operatorname{sd}v\operatorname{cd}v}{1 k^{2}{k^{\prime}}^% {2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E6 Encodings: TeX, pMML, png See also: Annotations for §22.8(i), §22.8 and Ch.22
22.8.7	$\displaystyle\operatorname{dc}(u v)$	$\displaystyle=\frac{\operatorname{dc}u\operatorname{dc}v {k^{\prime}}^{2}% \operatorname{sc}u\operatorname{nc}u\operatorname{sc}v\operatorname{nc}v}{1-{k% ^{\prime}}^{2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$
ⓘ Symbols: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E7 Encodings: TeX, pMML, png See also: Annotations for §22.8(i), §22.8 and Ch.22
22.8.8	$\displaystyle\operatorname{nc}(u v)$	$\displaystyle=\frac{\operatorname{nc}u\operatorname{nc}v \operatorname{sc}u% \operatorname{dc}u\operatorname{sc}v\operatorname{dc}v}{1-{k^{\prime}}^{2}{% \operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$
ⓘ Symbols: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E8 Encodings: TeX, pMML, png See also: Annotations for §22.8(i), §22.8 and Ch.22
22.8.9	$\displaystyle\operatorname{sc}(u v)$	$\displaystyle=\frac{\operatorname{sc}u\operatorname{dc}v\operatorname{nc}v % \operatorname{sc}v\operatorname{dc}u\operatorname{nc}u}{1-{k^{\prime}}^{2}{% \operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$
ⓘ Symbols: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E9 Encodings: TeX, pMML, png See also: Annotations for §22.8(i), §22.8 and Ch.22
22.8.10	$\displaystyle\operatorname{ns}(u v)$	$\displaystyle=\frac{\operatorname{ns}u\operatorname{ds}v\operatorname{cs}v-% \operatorname{ns}v\operatorname{ds}u\operatorname{cs}u}{{\operatorname{cs}}^{2% }v-{\operatorname{cs}}^{2}u},$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E10 Encodings: TeX, pMML, png See also: Annotations for §22.8(i), §22.8 and Ch.22
22.8.11	$\displaystyle\operatorname{ds}(u v)$	$\displaystyle=\frac{\operatorname{ds}u\operatorname{cs}v\operatorname{ns}v-% \operatorname{ds}v\operatorname{cs}u\operatorname{ns}u}{{\operatorname{cs}}^{2% }v-{\operatorname{cs}}^{2}u},$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E11 Encodings: TeX, pMML, png See also: Annotations for §22.8(i), §22.8 and Ch.22
22.8.12	$\displaystyle\operatorname{cs}(u v)$	$\displaystyle=\frac{\operatorname{cs}u\operatorname{ds}v\operatorname{ns}v-% \operatorname{cs}v\operatorname{ds}u\operatorname{ns}u}{{\operatorname{cs}}^{2% }v-{\operatorname{cs}}^{2}u}.$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E12 Encodings: TeX, pMML, png See also: Annotations for §22.8(i), §22.8 and Ch.22

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

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Notes:: See Lawden (1989, pp. 33–36, 43), Whittaker and Watson (1927, pp. 494–498).
Permalink:: http://dlmf.nist.gov/22.8.ii
See also:: Annotations for §22.8 and Ch.22

For $u,v\in\mathbb{C}$ , and with the common modulus $k$ suppressed:

22.8.13	$\displaystyle\operatorname{sn}(u v)$	$\displaystyle=\frac{{\operatorname{sn}}^{2}u-{\operatorname{sn}}^{2}v}{% \operatorname{sn}u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v% \operatorname{cn}u\operatorname{dn}u},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E13 Encodings: TeX, pMML, png See also: Annotations for §22.8(ii), §22.8 and Ch.22
22.8.14	$\displaystyle\operatorname{sn}(u v)$	$\displaystyle=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname{dn}v % \operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{cn}u% \operatorname{cn}v \operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E14 Encodings: TeX, pMML, png See also: Annotations for §22.8(ii), §22.8 and Ch.22
22.8.15	$\displaystyle\operatorname{cn}(u v)$	$\displaystyle=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname{dn}v-% \operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{sn}u% \operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E15 Encodings: TeX, pMML, png See also: Annotations for §22.8(ii), §22.8 and Ch.22

22.8.16	$\displaystyle\operatorname{cn}(u v)$	$\displaystyle=\frac{1-{\operatorname{sn}}^{2}u-{\operatorname{sn}}^{2}v k^{2}{% \operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v}{\operatorname{cn}u% \operatorname{cn}v \operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E16 Encodings: TeX, pMML, png See also: Annotations for §22.8(ii), §22.8 and Ch.22
22.8.17	$\displaystyle\operatorname{dn}(u v)$	$\displaystyle=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}u-% \operatorname{sn}v\operatorname{cn}u\operatorname{dn}v}{\operatorname{sn}u% \operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E17 Encodings: TeX, pMML, png See also: Annotations for §22.8(ii), §22.8 and Ch.22
22.8.18	$\displaystyle\operatorname{dn}(u v)$	$\displaystyle=\frac{\operatorname{cn}u\operatorname{dn}u\operatorname{cn}v% \operatorname{dn}v {k^{\prime}}^{2}\operatorname{sn}u\operatorname{sn}v}{% \operatorname{cn}u\operatorname{cn}v \operatorname{sn}u\operatorname{dn}u% \operatorname{sn}v\operatorname{dn}v}.$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E18 Encodings: TeX, pMML, png See also: Annotations for §22.8(ii), §22.8 and Ch.22

§22.8(iii) Special Relations Between Arguments

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Keywords:: Jacobian elliptic functions, addition theorems, poristic polygon constructions, poristic polygon constructions of Poncelet, special values of the variable
Notes:: See Whittaker and Watson (1927, p. 530).
Permalink:: http://dlmf.nist.gov/22.8.iii
See also:: Annotations for §22.8 and Ch.22

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 See also: Annotations for §22.8(iii), §22.8 and Ch.22

Then

22.8.20		$\begin{vmatrix}\operatorname{sn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}% z_{1}&1\\ \operatorname{sn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}&1\\ \operatorname{sn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}&1\\ \operatorname{sn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}&1\end{% vmatrix}=0,$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\det$ : determinant, $z$ : complex and $k$ : modulus Referenced by: §22.8(iii) Permalink: http://dlmf.nist.gov/22.8.E20 Encodings: TeX, pMML, png See also: Annotations for §22.8(iii), §22.8 and Ch.22

and

22.8.21		${k^{\prime}}^{2}-{k^{\prime}}^{2}k^{2}\operatorname{sn}z_{1}\operatorname{sn}z% _{2}\operatorname{sn}z_{3}\operatorname{sn}z_{4} k^{2}\operatorname{cn}z_{1}% \operatorname{cn}z_{2}\operatorname{cn}z_{3}\operatorname{cn}z_{4}-% \operatorname{dn}z_{1}\operatorname{dn}z_{2}\operatorname{dn}z_{3}% \operatorname{dn}z_{4}=0.$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.8.E21 Encodings: TeX, pMML, png See also: Annotations for §22.8(iii), §22.8 and Ch.22

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		$z_{1} z_{2} z_{3} z_{4}=2K\left(k\right).$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : 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 See also: Annotations for §22.8(iii), §22.8 and Ch.22

Then

22.8.23		$\begin{vmatrix}\operatorname{sn}z_{1}\operatorname{cn}z_{1}&\operatorname{cn}z% _{1}\operatorname{dn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}z_{1}\\ \operatorname{sn}z_{2}\operatorname{cn}z_{2}&\operatorname{cn}z_{2}% \operatorname{dn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}\\ \operatorname{sn}z_{3}\operatorname{cn}z_{3}&\operatorname{cn}z_{3}% \operatorname{dn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}\\ \operatorname{sn}z_{4}\operatorname{cn}z_{4}&\operatorname{cn}z_{4}% \operatorname{dn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}\end{% vmatrix}=0.$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\det$ : determinant, $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E23 Encodings: TeX, pMML, png See also: Annotations for §22.8(iii), §22.8 and Ch.22

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

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

22.8.24		$z_{1}-z_{2}=z_{2}-z_{3}=\tfrac{2}{3}K\left(k\right),$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : 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 See also: Annotations for §22.8(iii), §22.8 and Ch.22

then

22.8.25		$\frac{(\operatorname{dn}z_{2} \operatorname{dn}z_{3})(\operatorname{dn}z_{3} % \operatorname{dn}z_{1})(\operatorname{dn}z_{1} \operatorname{dn}z_{2})}{% \operatorname{dn}z_{1} \operatorname{dn}z_{2} \operatorname{dn}z_{3}}$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.8.E25 Encodings: TeX, pMML, png See also: Annotations for §22.8(iii), §22.8 and Ch.22

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

22.8.26		$z_{1}-z_{2}=z_{2}-z_{3}=z_{3}-z_{4}=\tfrac{1}{2}K\left(k\right),$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : 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 See also: Annotations for §22.8(iii), §22.8 and Ch.22

then

22.8.27		$\operatorname{dn}z_{1}\operatorname{dn}z_{3}=\operatorname{dn}z_{2}% \operatorname{dn}z_{4}=k^{\prime}.$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.8.E27 Encodings: TeX, pMML, png See also: Annotations for §22.8(iii), §22.8 and Ch.22

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

22.7 Landen Transformations 22.9 Cyclic Identities

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§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