§23.10 Addition Theorems and Other Identities

Permalink:: http://dlmf.nist.gov/23.10

§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

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

23.10.1 $\mathop{\wp\/}\nolimits\!\left(u+v\right)=\frac{1}{4}\left(\frac{{\mathop{\wp\/}\nolimits^{{\prime}}}\!\left(u\right)-{\mathop{\wp\/}\nolimits^{{\prime}}}\!\left(v\right)}{\mathop{\wp\/}\nolimits\!\left(u\right)-\mathop{\wp\/}\nolimits\!\left(v\right)}\right)^{2}-\mathop{\wp\/}\nolimits\!\left(u\right)-\mathop{\wp\/}\nolimits\!\left(v\right),$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -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

23.10.2 $\mathop{\zeta\/}\nolimits\!\left(u+v\right)=\mathop{\zeta\/}\nolimits\!\left(u\right)+\mathop{\zeta\/}\nolimits\!\left(v\right)+\frac{1}{2}\frac{{\mathop{\zeta\/}\nolimits^{{\prime\prime}}}\!\left(u\right)-{\mathop{\zeta\/}\nolimits^{{\prime\prime}}}\!\left(v\right)}{{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(u\right)-{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(v\right)},$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): 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

23.10.3 $\frac{\mathop{\sigma\/}\nolimits\!\left(u+v\right)\mathop{\sigma\/}\nolimits\!\left(u-v\right)}{{\mathop{\sigma\/}\nolimits^{{2}}}\!\left(u\right){\mathop{\sigma\/}\nolimits^{{2}}}\!\left(v\right)}=\mathop{\wp\/}\nolimits\!\left(v\right)-\mathop{\wp\/}\nolimits\!\left(u\right),$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): 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

23.10.4 $\mathop{\sigma\/}\nolimits\!\left(u+v\right)\mathop{\sigma\/}\nolimits\!\left(u-v\right)\mathop{\sigma\/}\nolimits\!\left(x+y\right)\mathop{\sigma\/}\nolimits\!\left(x-y\right)+\mathop{\sigma\/}\nolimits\!\left(v+x\right)\mathop{\sigma\/}\nolimits\!\left(v-x\right)\mathop{\sigma\/}\nolimits\!\left(u+y\right)\mathop{\sigma\/}\nolimits\!\left(u-y\right)+{\mathop{\sigma\/}\nolimits\!\left(x+u\right)\mathop{\sigma\/}\nolimits\!\left(x-u\right)\mathop{\sigma\/}\nolimits\!\left(v+y\right)\mathop{\sigma\/}\nolimits\!\left(v-y\right)=0.}$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): 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

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

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathbb{L}$ : lattice and $w=-u-v$
Referenced by:: §22.8(iii)
Permalink:: http://dlmf.nist.gov/23.10.E5
Encodings:: TeX, pMML, png

and

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

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice and $w=-u-v$
Permalink:: http://dlmf.nist.gov/23.10.E6
Encodings:: TeX, pMML, png

§23.10(ii) 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).
Keywords:: Weierstrass elliptic functions
Permalink:: http://dlmf.nist.gov/23.10.ii

23.10.7 $\mathop{\wp\/}\nolimits\!\left(2z\right)=-2\mathop{\wp\/}\nolimits\!\left(z\right)+\frac{1}{4}\left(\frac{{\mathop{\wp\/}\nolimits^{{\prime\prime}}}\!\left(z\right)}{{\mathop{\wp\/}\nolimits^{{\prime}}}\!\left(z\right)}\right)^{2},$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -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

23.10.8 $(\mathop{\wp\/}\nolimits\!\left(2z\right)-e_{1}){{\mathop{\wp\/}\nolimits^{{\prime}}}}^{2}(z)=\left((\mathop{\wp\/}\nolimits\!\left(z\right)-e_{1})^{2}-(e_{1}-e_{2})(e_{1}-e_{3})\right)^{2}.$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathbb{L}$ : lattice, $e_{j}$ : zeros and $z$ : complex
Referenced by:: §23.10(ii), §23.10(ii)
Permalink:: http://dlmf.nist.gov/23.10.E8
Encodings:: TeX, pMML, png

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

23.10.9 $\mathop{\zeta\/}\nolimits\!\left(2z\right)=2\mathop{\zeta\/}\nolimits\!\left(z\right)+\frac{1}{2}\frac{{\mathop{\zeta\/}\nolimits^{{\prime\prime\prime}}}\!\left(z\right)}{{\mathop{\zeta\/}\nolimits^{{\prime\prime}}}\!\left(z\right)},$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): 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

23.10.10 $\mathop{\sigma\/}\nolimits\!\left(2z\right)=-{\mathop{\wp\/}\nolimits^{{\prime}}}\!\left(z\right){\mathop{\sigma\/}\nolimits^{{4}}}\!\left(z\right).$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): 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

§23.10(iii) $n$ -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).
Keywords:: Weierstrass elliptic functions
Permalink:: http://dlmf.nist.gov/23.10.iii

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

23.10.11 $n^{2}\mathop{\wp\/}\nolimits\!\left(nz\right)=\sum _{{j=0}}^{{n-1}}\sum _{{\ell=0}}^{{n-1}}\mathop{\wp\/}\nolimits\!\left(z+\frac{2j}{n}\omega _{1}+\frac{2\ell}{n}\omega _{3}\right),$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -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

23.10.12 $n\mathop{\zeta\/}\nolimits\!\left(nz\right)=-n(n-1)(\eta _{1}+\eta _{3})+\sum _{{j=0}}^{{n-1}}\sum _{{\ell=0}}^{{n-1}}\mathop{\zeta\/}\nolimits\!\left(z+\frac{2j}{n}\omega _{1}+\frac{2\ell}{n}\omega _{3}\right),$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
Referenced by:: §23.10(iii)
Permalink:: http://dlmf.nist.gov/23.10.E12
Encodings:: TeX, pMML, png

23.10.13 $\mathop{\sigma\/}\nolimits\!\left(nz\right)=A_{n}e^{{-n(n-1)(\eta _{1}+\eta _{3})z}}\prod _{{j=0}}^{{n-1}}\prod _{{\ell=0}}^{{n-1}}\mathop{\sigma\/}\nolimits\!\left(z+\frac{2j}{n}\omega _{1}+\frac{2\ell}{n}\omega _{3}\right),$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $e$ : base of exponential function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $A_{n}$ : coefficients and $\eta _{j}$
Referenced by:: §23.10(iii)
Permalink:: http://dlmf.nist.gov/23.10.E13
Encodings:: TeX, pMML, png

where

23.10.14 $A_{n}=n\prod _{{j=0}}^{{n-1}}\prod _{{\substack{\ell=0\\ \ell\neq j}}}^{{n-1}}\frac{1}{\mathop{\sigma\/}\nolimits\!\left((2j\omega _{1}+2\ell\omega _{3})/n\right)}.$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): 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

Equivalently,

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