# §23.10 Addition Theorems and Other Identities

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 23.10.1 $\wp\left(u+v\right)=\frac{1}{4}\left(\frac{\wp'\left(u\right)-\wp'\left(v% \right)}{\wp\left(u\right)-\wp\left(v\right)}\right)^{2}-\wp\left(u\right)-\wp% \left(v\right),$ ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$): Weierstrass $\wp$-function and $\mathbb{L}$: lattice A&S Ref: 18.4.1 Referenced by: §23.10(ii), §23.20(ii), §23.20(ii), §23.23, §23.9 Permalink: http://dlmf.nist.gov/23.10.E1 Encodings: TeX, pMML, png See also: Annotations for §23.10(i), §23.10 and Ch.23
 23.10.2 $\zeta\left(u+v\right)=\zeta\left(u\right)+\zeta\left(v\right)+\frac{1}{2}\frac% {\zeta''\left(u\right)-\zeta''\left(v\right)}{\zeta'\left(u\right)-\zeta'\left% (v\right)},$ ⓘ Symbols: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$): 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 and Ch.23
 23.10.3 $\frac{\sigma\left(u+v\right)\sigma\left(u-v\right)}{{\sigma}^{2}\left(u\right)% {\sigma}^{2}\left(v\right)}=\wp\left(v\right)-\wp\left(u\right),$
 23.10.4 $\sigma\left(u+v\right)\sigma\left(u-v\right)\sigma\left(x+y\right)\sigma\left(% x-y\right)+\sigma\left(v+x\right)\sigma\left(v-x\right)\sigma\left(u+y\right)% \sigma\left(u-y\right)+{\sigma\left(x+u\right)\sigma\left(x-u\right)\sigma% \left(v+y\right)\sigma\left(v-y\right)=0.}$

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

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

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

and

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

## §23.10(ii) Duplication Formulas

 23.10.7 $\wp\left(2z\right)=-2\wp\left(z\right)+\frac{1}{4}\left(\frac{\wp''\left(z% \right)}{\wp'\left(z\right)}\right)^{2},$ ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$): Weierstrass $\wp$-function, $\mathbb{L}$: lattice and $z$: complex A&S Ref: 18.4.5 Referenced by: §23.10(ii) Permalink: http://dlmf.nist.gov/23.10.E7 Encodings: TeX, pMML, png See also: Annotations for §23.10(ii), §23.10 and Ch.23
 23.10.8 $(\wp\left(2z\right)-e_{1}){\wp'}^{2}(z)=\left((\wp\left(z\right)-e_{1})^{2}-(e% _{1}-e_{2})(e_{1}-e_{3})\right)^{2}.$

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

 23.10.9 $\zeta\left(2z\right)=2\zeta\left(z\right)+\frac{1}{2}\frac{\zeta'''\left(z% \right)}{\zeta''\left(z\right)},$ ⓘ Symbols: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$): Weierstrass zeta function, $\mathbb{L}$: lattice and $z$: complex A&S Ref: 18.4.7 Referenced by: §23.10(ii) Permalink: http://dlmf.nist.gov/23.10.E9 Encodings: TeX, pMML, png See also: Annotations for §23.10(ii), §23.10 and Ch.23
 23.10.10 $\sigma\left(2z\right)=-\wp'\left(z\right){\sigma}^{4}\left(z\right).$

## §23.10(iii) $n$-Tuple Formulas

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

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

where

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

Equivalently,

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

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

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).