# §22.8(i) Sum of Two Arguments

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

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

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

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

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

# §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

Then

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

and

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

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

Then

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

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

then

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

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

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

then

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

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