# §23.10 Addition Theorems and Other Identities

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 23.10.1 $\mathop{\wp\/}\nolimits\!\left(u+v\right)=\frac{1}{4}\left(\frac{\mathop{\wp\/% }\nolimits'\!\left(u\right)-\mathop{\wp\/}\nolimits'\!\left(v\right)}{\mathop{% \wp\/}\nolimits\!\left(u\right)-\mathop{\wp\/}\nolimits\!\left(v\right)}\right% )^{2}-\mathop{\wp\/}\nolimits\!\left(u\right)-\mathop{\wp\/}\nolimits\!\left(v% \right),$
 23.10.2 $\mathop{\zeta\/}\nolimits\!\left(u+v\right)=\mathop{\zeta\/}\nolimits\!\left(u% \right)+\mathop{\zeta\/}\nolimits\!\left(v\right)+\frac{1}{2}\frac{\mathop{% \zeta\/}\nolimits''\!\left(u\right)-\mathop{\zeta\/}\nolimits''\!\left(v\right% )}{\mathop{\zeta\/}\nolimits'\!\left(u\right)-\mathop{\zeta\/}\nolimits'\!% \left(v\right)},$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{z}\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$): Weierstrass zeta function and $\mathbb{L}$: lattice A&S Ref: 18.4.3 Referenced by: §23.9 Permalink: http://dlmf.nist.gov/23.10.E2 Encodings: TeX, pMML, png See also: Annotations for 23.10(i)
 23.10.3 $\frac{\mathop{\sigma\/}\nolimits\!\left(u+v\right)\mathop{\sigma\/}\nolimits\!% \left(u-v\right)}{{\mathop{\sigma\/}\nolimits^{2}}\!\left(u\right){\mathop{% \sigma\/}\nolimits^{2}}\!\left(v\right)}=\mathop{\wp\/}\nolimits\!\left(v% \right)-\mathop{\wp\/}\nolimits\!\left(u\right),$
 23.10.4 $\mathop{\sigma\/}\nolimits\!\left(u+v\right)\mathop{\sigma\/}\nolimits\!\left(% u-v\right)\mathop{\sigma\/}\nolimits\!\left(x+y\right)\mathop{\sigma\/}% \nolimits\!\left(x-y\right)+\mathop{\sigma\/}\nolimits\!\left(v+x\right)% \mathop{\sigma\/}\nolimits\!\left(v-x\right)\mathop{\sigma\/}\nolimits\!\left(% u+y\right)\mathop{\sigma\/}\nolimits\!\left(u-y\right)+{\mathop{\sigma\/}% \nolimits\!\left(x+u\right)\mathop{\sigma\/}\nolimits\!\left(x-u\right)\mathop% {\sigma\/}\nolimits\!\left(v+y\right)\mathop{\sigma\/}\nolimits\!\left(v-y% \right)=0.}$

For further addition-type identities for the $\mathop{\sigma\/}\nolimits$-function see Lawden (1989, §6.4).

If $u+v+w=0$, then

 23.10.5 $\begin{vmatrix}1&\mathop{\wp\/}\nolimits\!\left(u\right)&\mathop{\wp\/}% \nolimits'\!\left(u\right)\\ 1&\mathop{\wp\/}\nolimits\!\left(v\right)&\mathop{\wp\/}\nolimits'\!\left(v% \right)\\ 1&\mathop{\wp\/}\nolimits\!\left(w\right)&\mathop{\wp\/}\nolimits'\!\left(w% \right)\end{vmatrix}=0,$

and

 23.10.6 $\left(\mathop{\zeta\/}\nolimits\!\left(u\right)+\mathop{\zeta\/}\nolimits\!% \left(v\right)+\mathop{\zeta\/}\nolimits\!\left(w\right)\right)^{2}+\mathop{% \zeta\/}\nolimits'\!\left(u\right)+\mathop{\zeta\/}\nolimits'\!\left(v\right)+% \mathop{\zeta\/}\nolimits'\!\left(w\right)=0.$

## §23.10(ii) Duplication Formulas

 23.10.7 $\mathop{\wp\/}\nolimits\!\left(2z\right)=-2\mathop{\wp\/}\nolimits\!\left(z% \right)+\frac{1}{4}\left(\frac{\mathop{\wp\/}\nolimits''\!\left(z\right)}{% \mathop{\wp\/}\nolimits'\!\left(z\right)}\right)^{2},$
 23.10.8 $(\mathop{\wp\/}\nolimits\!\left(2z\right)-e_{1}){\mathop{\wp\/}\nolimits'}^{2}% (z)=\left((\mathop{\wp\/}\nolimits\!\left(z\right)-e_{1})^{2}-(e_{1}-e_{2})(e_% {1}-e_{3})\right)^{2}.$

(23.10.8) continues to hold when $e_{1}$, $e_{2}$, $e_{3}$ are permuted cyclically.

 23.10.9 $\mathop{\zeta\/}\nolimits\!\left(2z\right)=2\mathop{\zeta\/}\nolimits\!\left(z% \right)+\frac{1}{2}\frac{\mathop{\zeta\/}\nolimits'''\!\left(z\right)}{\mathop% {\zeta\/}\nolimits''\!\left(z\right)},$
 23.10.10 $\mathop{\sigma\/}\nolimits\!\left(2z\right)=-\mathop{\wp\/}\nolimits'\!\left(z% \right){\mathop{\sigma\/}\nolimits^{4}}\!\left(z\right).$

## §23.10(iii) $n$-Tuple Formulas

For $n=2,3,\dots$,

 23.10.11 $n^{2}\mathop{\wp\/}\nolimits\!\left(nz\right)=\sum_{j=0}^{n-1}\sum_{\ell=0}^{n% -1}\mathop{\wp\/}\nolimits\!\left(z+\frac{2j}{n}\omega_{1}+\frac{2\ell}{n}% \omega_{3}\right),$
 23.10.12 $n\mathop{\zeta\/}\nolimits\!\left(nz\right)=-n(n-1)(\eta_{1}+\eta_{3})+\sum_{j% =0}^{n-1}\sum_{\ell=0}^{n-1}\mathop{\zeta\/}\nolimits\!\left(z+\frac{2j}{n}% \omega_{1}+\frac{2\ell}{n}\omega_{3}\right),$
 23.10.13 $\mathop{\sigma\/}\nolimits\!\left(nz\right)=A_{n}e^{-n(n-1)(\eta_{1}+\eta_{3})% z}\prod_{j=0}^{n-1}\prod_{\ell=0}^{n-1}\mathop{\sigma\/}\nolimits\!\left(z+% \frac{2j}{n}\omega_{1}+\frac{2\ell}{n}\omega_{3}\right),$

where

 23.10.14 $A_{n}=n\prod_{j=0}^{n-1}\prod_{\begin{subarray}{c}\ell=0\\ \ell\neq j\end{subarray}}^{n-1}\frac{1}{\mathop{\sigma\/}\nolimits\!\left((2j% \omega_{1}+2\ell\omega_{3})/n\right)}.$

Equivalently,

 23.10.15 $A_{n}=\left(\frac{\pi^{2}G^{2}}{\omega_{1}}\right)^{n^{2}-1}\frac{q^{n(n-1)/2}% }{i^{n-1}}\mathop{\exp\/}\nolimits\!\left(-\frac{(n-1)\eta_{1}}{3\omega_{1}}% \left((2n-1)(\omega_{1}^{2}+\omega_{3}^{2})+3(n-1)\omega_{1}\omega_{3}\right)% \right),$

where

 23.10.16 $\displaystyle q$ $\displaystyle=e^{\pi i\omega_{3}/\omega_{1}},$ $\displaystyle G$ $\displaystyle=\prod_{n=1}^{\infty}(1-q^{2n}).$

## §23.10(iv) Homogeneity

For any nonzero real or complex constant $c$

 23.10.17 $\displaystyle\mathop{\wp\/}\nolimits\!\left(cz|c\mathbb{L}\right)$ $\displaystyle=c^{-2}\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right),$ 23.10.18 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(cz|c\mathbb{L}\right)$ $\displaystyle=c^{-1}\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right),$ 23.10.19 $\displaystyle\mathop{\sigma\/}\nolimits\!\left(cz|c\mathbb{L}\right)$ $\displaystyle=c\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right).$

Also, when $\mathbb{L}$ is replaced by $c\mathbb{L}$ the lattice invariants $g_{2}$ and $g_{3}$ are divided by $c^{4}$ and $c^{6}$, respectively.

For these results and further identities see Lawden (1989, §6.6) and Apostol (1990, p. 14).