§23.10 Addition Theorems and Other Identities

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Permalink:: http://dlmf.nist.gov/23.10
See also:: Annotations for Ch.23

§23.10(i) Addition Theorems
§23.10(ii) Duplication Formulas
§23.10(iii) $n$ -Tuple Formulas
§23.10(iv) Homogeneity

§23.10(i) Addition Theorems

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Keywords:: Weierstrass elliptic functions, addition theorems
Notes:: See Whittaker and Watson (1927, §§20.3–20.311, 20.41), Lawden (1989, pp. 152–154, 158, 161–162).
Permalink:: http://dlmf.nist.gov/23.10.i
See also:: Annotations for §23.10 and Ch.23

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		$\frac{\sigma\left(u+v\right)\sigma\left(u-v\right)}{{\sigma}^{2}\left(u\right)% {\sigma}^{2}\left(v\right)}=\wp\left(v\right)-\wp\left(u\right),$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function and $\mathbb{L}$ : lattice A&S Ref: 18.4.4 Referenced by: §23.10(ii) Permalink: http://dlmf.nist.gov/23.10.E3 Encodings: TeX, pMML, png See also: Annotations for §23.10(i), §23.10 and Ch.23

23.10.4		$\sigma\left(u+v\right)\sigma\left(u-v\right)\sigma\left(x+y\right)\sigma\left(% x-y\right)+\sigma\left(v+x\right)\sigma\left(v-x\right)\sigma\left(u+y\right)% \sigma\left(u-y\right)+{\sigma\left(x+u\right)\sigma\left(x-u\right)\sigma% \left(v+y\right)\sigma\left(v-y\right)=0.}$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice, $x$ : real part of $z$ and $y$ : imaginary part of $z$ Permalink: http://dlmf.nist.gov/23.10.E4 Encodings: TeX, pMML, png See also: Annotations for §23.10(i), §23.10 and Ch.23

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

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

23.10.5		$\begin{vmatrix}1&\wp\left(u\right)&\wp'\left(u\right)\\ 1&\wp\left(v\right)&\wp'\left(v\right)\\ 1&\wp\left(w\right)&\wp'\left(w\right)\end{vmatrix}=0,$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\det$ : determinant, $\mathbb{L}$ : lattice and $w=-u-v$ Referenced by: §22.8(iii) Permalink: http://dlmf.nist.gov/23.10.E5 Encodings: TeX, pMML, png See also: Annotations for §23.10(i), §23.10 and Ch.23

and

23.10.6		$\left(\zeta\left(u\right)+\zeta\left(v\right)+\zeta\left(w\right)\right)^{2}+% \zeta'\left(u\right)+\zeta'\left(v\right)+\zeta'\left(w\right)=0.$
ⓘ 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 $w=-u-v$ Permalink: http://dlmf.nist.gov/23.10.E6 Encodings: TeX, pMML, png See also: Annotations for §23.10(i), §23.10 and Ch.23

§23.10(ii) Duplication Formulas

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Keywords:: Weierstrass elliptic functions, duplication formulas
Notes:: For (23.10.7), (23.10.9), (23.10.10) let $v\to u$ in (23.10.1)–(23.10.3). For (23.10.8) see Walker (1996, p. 83).
Permalink:: http://dlmf.nist.gov/23.10.ii
See also:: Annotations for §23.10 and Ch.23

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		$(\wp\left(2z\right)-e_{1}){\wp'}^{2}(z)=\left((\wp\left(z\right)-e_{1})^{2}-(e% _{1}-e_{2})(e_{1}-e_{3})\right)^{2}.$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice and $z$ : complex Referenced by: §23.10(ii), §23.10(ii) Permalink: http://dlmf.nist.gov/23.10.E8 Encodings: TeX, pMML, png See also: Annotations for §23.10(ii), §23.10 and Ch.23

(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		$\sigma\left(2z\right)=-\wp'\left(z\right){\sigma}^{4}\left(z\right).$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice and $z$ : complex A&S Ref: 18.4.8 Referenced by: §23.10(ii) Permalink: http://dlmf.nist.gov/23.10.E10 Encodings: TeX, pMML, png See also: Annotations for §23.10(ii), §23.10 and Ch.23

§23.10(iii) $n$ -Tuple Formulas

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Keywords:: Weierstrass elliptic functions, $n$ -tuple formulas
Notes:: See Lawden (1989, §6.6). For (23.10.11) and (23.10.12) compare the poles and residues of the two sides. (23.10.13) follows by integration. For (23.10.15) combine (23.10.14), (23.8.7), and (4.21.35).
Permalink:: http://dlmf.nist.gov/23.10.iii
See also:: Annotations for §23.10 and Ch.23

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

23.10.11		$n^{2}\wp\left(nz\right)=\sum_{j=0}^{n-1}\sum_{\ell=0}^{n-1}\wp\left(z+\frac{2j% }{n}\omega_{1}+\frac{2\ell}{n}\omega_{3}\right),$
ⓘ 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, $n$ : integer, $z$ : complex and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators Referenced by: §23.10(iii) Permalink: http://dlmf.nist.gov/23.10.E11 Encodings: TeX, pMML, png See also: Annotations for §23.10(iii), §23.10 and Ch.23

23.10.12		$n\zeta\left(nz\right)=-n(n-1)(\eta_{1}+\eta_{3})+\sum_{j=0}^{n-1}\sum_{\ell=0}% ^{n-1}\zeta\left(z+\frac{2j}{n}\omega_{1}+\frac{2\ell}{n}\omega_{3}\right),$
ⓘ 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, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Referenced by: §23.10(iii) Permalink: http://dlmf.nist.gov/23.10.E12 Encodings: TeX, pMML, png See also: Annotations for §23.10(iii), §23.10 and Ch.23

23.10.13		$\sigma\left(nz\right)=A_{n}e^{-n(n-1)(\eta_{1}+\eta_{3})z}\prod_{j=0}^{n-1}% \prod_{\ell=0}^{n-1}\sigma\left(z+\frac{2j}{n}\omega_{1}+\frac{2\ell}{n}\omega% _{3}\right),$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathrm{e}$ : base of natural logarithm, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $A_{n}$ : coefficients and $\eta_{j}$ : complex number Referenced by: §23.10(iii) Permalink: http://dlmf.nist.gov/23.10.E13 Encodings: TeX, pMML, png See also: Annotations for §23.10(iii), §23.10 and Ch.23

where

23.10.14		$A_{n}=n\prod_{j=0}^{n-1}\prod_{\begin{subarray}{c}\ell=0\\ \ell\neq j\end{subarray}}^{n-1}\frac{1}{\sigma\left((2j\omega_{1}+2\ell\omega_% {3})/n\right)}.$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice, $n$ : integer, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $A_{n}$ : coefficients Referenced by: §23.10(iii) Permalink: http://dlmf.nist.gov/23.10.E14 Encodings: TeX, pMML, png See also: Annotations for §23.10(iii), §23.10 and Ch.23

Equivalently,

23.10.15		$A_{n}=\left(\frac{\pi^{2}G^{2}}{\omega_{1}}\right)^{n^{2}-1}\frac{q^{n(n-1)/2}% }{i^{n-1}}\exp\left(-\frac{(n-1)\eta_{1}}{3\omega_{1}}\left((2n-1)(\omega_{1}^% {2}+\omega_{3}^{2})+3(n-1)\omega_{1}\omega_{3}\right)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $q$ : nome, $n$ : integer, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $A_{n}$ : coefficients, $G$ and $\eta_{j}$ : complex number Referenced by: §23.10(iii) Permalink: http://dlmf.nist.gov/23.10.E15 Encodings: TeX, pMML, png See also: Annotations for §23.10(iii), §23.10 and Ch.23

where

23.10.16	$\displaystyle q$	$\displaystyle=e^{\pi i\omega_{3}/\omega_{1}},$
23.10.16	$\displaystyle G$	$\displaystyle=\prod_{n=1}^{\infty}(1-q^{2n}).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $q$ : nome, $n$ : integer, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $G$ Permalink: http://dlmf.nist.gov/23.10.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.10(iii), §23.10 and Ch.23

§23.10(iv) Homogeneity

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Keywords:: Weierstrass elliptic functions, homogeneity
Referenced by:: §23.1
Permalink:: http://dlmf.nist.gov/23.10.iv
See also:: Annotations for §23.10 and Ch.23

For any nonzero real or complex constant $c$

23.10.17	$\displaystyle\wp\left(cz\|c\mathbb{L}\right)$	$\displaystyle=c^{-2}\wp\left(z\|\mathbb{L}\right),$
ⓘ 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, $z$ : complex and $c$ : constant A&S Ref: 18.2.2 Permalink: http://dlmf.nist.gov/23.10.E17 Encodings: TeX, pMML, png See also: Annotations for §23.10(iv), §23.10 and Ch.23
23.10.18	$\displaystyle\zeta\left(cz\|c\mathbb{L}\right)$	$\displaystyle=c^{-1}\zeta\left(z\|\mathbb{L}\right),$
ⓘ 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, $z$ : complex and $c$ : constant A&S Ref: 18.2.3 Permalink: http://dlmf.nist.gov/23.10.E18 Encodings: TeX, pMML, png See also: Annotations for §23.10(iv), §23.10 and Ch.23
23.10.19	$\displaystyle\sigma\left(cz\|c\mathbb{L}\right)$	$\displaystyle=c\sigma\left(z\|\mathbb{L}\right).$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice, $z$ : complex and $c$ : constant A&S Ref: 18.2.4 Permalink: http://dlmf.nist.gov/23.10.E19 Encodings: TeX, pMML, png See also: Annotations for §23.10(iv), §23.10 and Ch.23

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).

23.10.17	$\displaystyle\wp\left(cz\|c\mathbb{L}\right)$	$\displaystyle=c^{-2}\wp\left(z\|\mathbb{L}\right),$
ⓘ 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, $z$ : complex and $c$ : constant A&S Ref: 18.2.2 Permalink: http://dlmf.nist.gov/23.10.E17 Encodings: TeX, pMML, png See also: Annotations for §23.10(iv), §23.10 and Ch.23
23.10.18	$\displaystyle\zeta\left(cz\|c\mathbb{L}\right)$	$\displaystyle=c^{-1}\zeta\left(z\|\mathbb{L}\right),$
ⓘ 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, $z$ : complex and $c$ : constant A&S Ref: 18.2.3 Permalink: http://dlmf.nist.gov/23.10.E18 Encodings: TeX, pMML, png See also: Annotations for §23.10(iv), §23.10 and Ch.23
23.10.19	$\displaystyle\sigma\left(cz\|c\mathbb{L}\right)$	$\displaystyle=c\sigma\left(z\|\mathbb{L}\right).$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice, $z$ : complex and $c$ : constant A&S Ref: 18.2.4 Permalink: http://dlmf.nist.gov/23.10.E19 Encodings: TeX, pMML, png See also: Annotations for §23.10(iv), §23.10 and Ch.23

§23.10 Addition Theorems and Other Identities

Contents

§23.10(i) Addition Theorems

§23.10(ii) Duplication Formulas

§23.10(iii) n-Tuple Formulas

§23.10(iv) Homogeneity

§23.10(iii) $n$ -Tuple Formulas