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

§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 suppressed:

22.8.1 $\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}{1-k^{2}{% \operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v},$

ⓘ

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, : complex, : complex and : 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 $\operatorname{cn}(u+v)=\frac{\operatorname{cn}u\operatorname{cn}v-% \operatorname{sn}u\operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}{1-k^% {2}{\operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v},$

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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, : complex, : complex and : 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 $\operatorname{dn}(u+v)=\frac{\operatorname{dn}u\operatorname{dn}v-k^{2}% \operatorname{sn}u\operatorname{cn}u\operatorname{sn}v\operatorname{cn}v}{1-k^% {2}{\operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v}.$

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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, : complex, : complex and : 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 $\operatorname{cd}(u+v)=\frac{\operatorname{cd}u\operatorname{cd}v-{k^{\prime}}% ^{2}\operatorname{sd}u\operatorname{nd}u\operatorname{sd}v\operatorname{nd}v}{% 1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$

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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, : complex, : complex and : 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 $\operatorname{sd}(u+v)=\frac{\operatorname{sd}u\operatorname{cd}v\operatorname% {nd}v+\operatorname{sd}v\operatorname{cd}u\operatorname{nd}u}{1+k^{2}{k^{% \prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$

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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, : complex, : complex and : 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 $\operatorname{nd}(u+v)=\frac{\operatorname{nd}u\operatorname{nd}v+k^{2}% \operatorname{sd}u\operatorname{cd}u\operatorname{sd}v\operatorname{cd}v}{1+k^% {2}{k^{\prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$

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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, : complex, : complex and : 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 $\operatorname{dc}(u+v)=\frac{\operatorname{dc}u\operatorname{dc}v+{k^{\prime}}% ^{2}\operatorname{sc}u\operatorname{nc}u\operatorname{sc}v\operatorname{nc}v}{% 1-{k^{\prime}}^{2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$

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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, : complex, : complex and : 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 $\operatorname{nc}(u+v)=\frac{\operatorname{nc}u\operatorname{nc}v+% \operatorname{sc}u\operatorname{dc}u\operatorname{sc}v\operatorname{dc}v}{1-{k% ^{\prime}}^{2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$

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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, : complex, : complex and : 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 $\operatorname{sc}(u+v)=\frac{\operatorname{sc}u\operatorname{dc}v\operatorname% {nc}v+\operatorname{sc}v\operatorname{dc}u\operatorname{nc}u}{1-{k^{\prime}}^{% 2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$

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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, : complex, : complex and : 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 $\operatorname{ns}(u+v)=\frac{\operatorname{ns}u\operatorname{ds}v\operatorname% {cs}v-\operatorname{ns}v\operatorname{ds}u\operatorname{cs}u}{{\operatorname{% cs}}^{2}v-{\operatorname{cs}}^{2}u},$

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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, : complex, : complex and : 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 $\operatorname{ds}(u+v)=\frac{\operatorname{ds}u\operatorname{cs}v\operatorname% {ns}v-\operatorname{ds}v\operatorname{cs}u\operatorname{ns}u}{{\operatorname{% cs}}^{2}v-{\operatorname{cs}}^{2}u},$

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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, : complex, : complex and : 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 $\operatorname{cs}(u+v)=\frac{\operatorname{cs}u\operatorname{ds}v\operatorname% {ns}v-\operatorname{cs}v\operatorname{ds}u\operatorname{ns}u}{{\operatorname{% cs}}^{2}v-{\operatorname{cs}}^{2}u}.$

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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, : complex, : complex and : 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 suppressed:

22.8.13 $\operatorname{sn}(u+v)=\frac{{\operatorname{sn}}^{2}u-{\operatorname{sn}}^{2}v% }{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v% \operatorname{cn}u\operatorname{dn}u},$

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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, : complex, : complex and : 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 $\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{cn% }u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$

ⓘ

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, : complex, : complex and : 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 $\operatorname{cn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname% {dn}v-\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{sn% }u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},$

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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, : complex, : complex and : 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 $\operatorname{cn}(u+v)=\frac{1-{\operatorname{sn}}^{2}u-{\operatorname{sn}}^{2% }v+k^{2}{\operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v}{\operatorname{cn}u% \operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$

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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, : complex, : complex and : 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 $\operatorname{dn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname% {dn}u-\operatorname{sn}v\operatorname{cn}u\operatorname{dn}v}{\operatorname{sn% }u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},$

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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, : complex, : complex and : 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 $\operatorname{dn}(u+v)=\frac{\operatorname{cn}u\operatorname{dn}u\operatorname% {cn}v\operatorname{dn}v+{k^{\prime}}^{2}\operatorname{sn}u\operatorname{sn}v}{% \operatorname{cn}u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u% \operatorname{sn}v\operatorname{dn}v}.$

ⓘ

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, : complex, : complex and : 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 is again suppressed.

Let

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

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Symbols:: : 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, : complex and : 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, : complex, : 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 $z_{1}+z_{2}+z_{3}+z_{4}=2K\left(k\right).$

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : complex and : 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, : complex and : 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, : complex and : 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, : complex and : 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, : complex and : 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, : complex, : 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