# §22.8 Addition Theorems

## §22.8(i) Sum of Two Arguments

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

 22.8.1 $\displaystyle\operatorname{sn}(u+v)$ $\displaystyle=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}v+% \operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}{1-k^{2}{\operatorname{% sn}}^{2}u{\operatorname{sn}}^{2}v},$ 22.8.2 $\displaystyle\operatorname{cn}(u+v)$ $\displaystyle=\frac{\operatorname{cn}u\operatorname{cn}v-\operatorname{sn}u% \operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}{1-k^{2}{\operatorname{% sn}}^{2}u{\operatorname{sn}}^{2}v},$ 22.8.3 $\displaystyle\operatorname{dn}(u+v)$ $\displaystyle=\frac{\operatorname{dn}u\operatorname{dn}v-k^{2}\operatorname{sn% }u\operatorname{cn}u\operatorname{sn}v\operatorname{cn}v}{1-k^{2}{% \operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v}.$
 22.8.4 $\operatorname{cd}(u+v)=\frac{\operatorname{cd}u\operatorname{cd}v-{k^{\prime}}% ^{2}\operatorname{sd}u\operatorname{nd}u\operatorname{sd}v\operatorname{nd}v}{% 1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$
 22.8.5 $\displaystyle\operatorname{sd}(u+v)$ $\displaystyle=\frac{\operatorname{sd}u\operatorname{cd}v\operatorname{nd}v+% \operatorname{sd}v\operatorname{cd}u\operatorname{nd}u}{1+k^{2}{k^{\prime}}^{2% }{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$ 22.8.6 $\displaystyle\operatorname{nd}(u+v)$ $\displaystyle=\frac{\operatorname{nd}u\operatorname{nd}v+k^{2}\operatorname{sd% }u\operatorname{cd}u\operatorname{sd}v\operatorname{cd}v}{1+k^{2}{k^{\prime}}^% {2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$ 22.8.7 $\displaystyle\operatorname{dc}(u+v)$ $\displaystyle=\frac{\operatorname{dc}u\operatorname{dc}v+{k^{\prime}}^{2}% \operatorname{sc}u\operatorname{nc}u\operatorname{sc}v\operatorname{nc}v}{1-{k% ^{\prime}}^{2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$ 22.8.8 $\displaystyle\operatorname{nc}(u+v)$ $\displaystyle=\frac{\operatorname{nc}u\operatorname{nc}v+\operatorname{sc}u% \operatorname{dc}u\operatorname{sc}v\operatorname{dc}v}{1-{k^{\prime}}^{2}{% \operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$ 22.8.9 $\displaystyle\operatorname{sc}(u+v)$ $\displaystyle=\frac{\operatorname{sc}u\operatorname{dc}v\operatorname{nc}v+% \operatorname{sc}v\operatorname{dc}u\operatorname{nc}u}{1-{k^{\prime}}^{2}{% \operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$ 22.8.10 $\displaystyle\operatorname{ns}(u+v)$ $\displaystyle=\frac{\operatorname{ns}u\operatorname{ds}v\operatorname{cs}v-% \operatorname{ns}v\operatorname{ds}u\operatorname{cs}u}{{\operatorname{cs}}^{2% }v-{\operatorname{cs}}^{2}u},$ 22.8.11 $\displaystyle\operatorname{ds}(u+v)$ $\displaystyle=\frac{\operatorname{ds}u\operatorname{cs}v\operatorname{ns}v-% \operatorname{ds}v\operatorname{cs}u\operatorname{ns}u}{{\operatorname{cs}}^{2% }v-{\operatorname{cs}}^{2}u},$ 22.8.12 $\displaystyle\operatorname{cs}(u+v)$ $\displaystyle=\frac{\operatorname{cs}u\operatorname{ds}v\operatorname{ns}v-% \operatorname{cs}v\operatorname{ds}u\operatorname{ns}u}{{\operatorname{cs}}^{2% }v-{\operatorname{cs}}^{2}u}.$

See also Carlson (2004).

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

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

 22.8.13 $\displaystyle\operatorname{sn}(u+v)$ $\displaystyle=\frac{{\operatorname{sn}}^{2}u-{\operatorname{sn}}^{2}v}{% \operatorname{sn}u\operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v% \operatorname{cn}u\operatorname{dn}u},$ 22.8.14 $\displaystyle\operatorname{sn}(u+v)$ $\displaystyle=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname{dn}v+% \operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{cn}u% \operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$ 22.8.15 $\displaystyle\operatorname{cn}(u+v)$ $\displaystyle=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname{dn}v-% \operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{sn}u% \operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},$
 22.8.16 $\displaystyle\operatorname{cn}(u+v)$ $\displaystyle=\frac{1-{\operatorname{sn}}^{2}u-{\operatorname{sn}}^{2}v+k^{2}{% \operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v}{\operatorname{cn}u% \operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$ 22.8.17 $\displaystyle\operatorname{dn}(u+v)$ $\displaystyle=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}u-% \operatorname{sn}v\operatorname{cn}u\operatorname{dn}v}{\operatorname{sn}u% \operatorname{cn}v\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u},$ 22.8.18 $\displaystyle\operatorname{dn}(u+v)$ $\displaystyle=\frac{\operatorname{cn}u\operatorname{dn}u\operatorname{cn}v% \operatorname{dn}v+{k^{\prime}}^{2}\operatorname{sn}u\operatorname{sn}v}{% \operatorname{cn}u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u% \operatorname{sn}v\operatorname{dn}v}.$

See also Carlson (2004).

## §22.8(iii) Special Relations Between Arguments

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 $\begin{vmatrix}\operatorname{sn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}% z_{1}&1\\ \operatorname{sn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}&1\\ \operatorname{sn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}&1\\ \operatorname{sn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}&1\end{% vmatrix}=0,$

and

 22.8.21 ${k^{\prime}}^{2}-{k^{\prime}}^{2}k^{2}\operatorname{sn}z_{1}\operatorname{sn}z% _{2}\operatorname{sn}z_{3}\operatorname{sn}z_{4}+k^{2}\operatorname{cn}z_{1}% \operatorname{cn}z_{2}\operatorname{cn}z_{3}\operatorname{cn}z_{4}-% \operatorname{dn}z_{1}\operatorname{dn}z_{2}\operatorname{dn}z_{3}% \operatorname{dn}z_{4}=0.$

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 $\begin{vmatrix}\operatorname{sn}z_{1}\operatorname{cn}z_{1}&\operatorname{cn}z% _{1}\operatorname{dn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}z_{1}\\ \operatorname{sn}z_{2}\operatorname{cn}z_{2}&\operatorname{cn}z_{2}% \operatorname{dn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}\\ \operatorname{sn}z_{3}\operatorname{cn}z_{3}&\operatorname{cn}z_{3}% \operatorname{dn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}\\ \operatorname{sn}z_{4}\operatorname{cn}z_{4}&\operatorname{cn}z_{4}% \operatorname{dn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}\end{% vmatrix}=0.$

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 $\operatorname{dn}z_{1}\operatorname{dn}z_{3}=\operatorname{dn}z_{2}% \operatorname{dn}z_{4}=k^{\prime}.$ ⓘ

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