23 Weierstrass Elliptic and Modular Functions Weierstrass Elliptic Functions 23.5 Special Lattices 23.7 Quarter Periods

§23.6 Relations to Other Functions

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Permalink:: http://dlmf.nist.gov/23.6
See also:: Annotations for Ch.23

§23.6(i) Theta Functions
§23.6(ii) Jacobian Elliptic Functions
§23.6(iii) General Elliptic Functions
§23.6(iv) Elliptic Integrals

§23.6(i) Theta Functions

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Keywords:: Weierstrass elliptic functions, relations to other functions, theta functions, theta functions
Notes:: For (23.6.2)–(23.6.7) see Walker (1996, pp. 94 and 103). For (23.6.8) and (23.6.9) see Whittaker and Watson (1927, §21.43). For (23.6.10)–(23.6.12) use (23.6.9). For (23.6.13) and (23.6.14) see Lawden (1989, §6.6). For (23.6.15) combine (20.2.6) and (23.6.9).
Referenced by:: §20.9(ii)
Permalink:: http://dlmf.nist.gov/23.6.i
See also:: Annotations for §23.6 and Ch.23

In this subsection $2\omega_{1}$ , $2\omega_{3}$ are any pair of generators of the lattice $\mathbb{L}$ , and the lattice roots $e_{1}$ , $e_{2}$ , $e_{3}$ are given by (23.3.9).

23.6.1	$\displaystyle q$	$\displaystyle=e^{i\pi\tau},$
23.6.1	$\displaystyle\tau$	$\displaystyle=\omega_{3}/\omega_{1}.$
ⓘ 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, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\tau$ : complex variable A&S Ref: 18.10.1, 18.10.2, and 18.9.7 Permalink: http://dlmf.nist.gov/23.6.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.6(i), §23.6 and Ch.23

23.6.2	$\displaystyle e_{1}$	$\displaystyle=\frac{\pi^{2}}{12\omega_{1}^{2}}\left({\theta_{2}}^{4}\left(0,q% \right)+2{\theta_{4}}^{4}\left(0,q\right)\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{L}$ : lattice, $q$ : nome and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators A&S Ref: 18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11 Referenced by: §23.22(ii), §23.22(i), §23.6(i), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E2 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23
23.6.3	$\displaystyle e_{2}$	$\displaystyle=\frac{\pi^{2}}{12\omega_{1}^{2}}\left({\theta_{2}}^{4}\left(0,q% \right)-{\theta_{4}}^{4}\left(0,q\right)\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{L}$ : lattice, $q$ : nome and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators A&S Ref: 18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11 Permalink: http://dlmf.nist.gov/23.6.E3 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23
23.6.4	$\displaystyle e_{3}$	$\displaystyle=-\frac{\pi^{2}}{12\omega_{1}^{2}}\left(2{\theta_{2}}^{4}\left(0,% q\right)+{\theta_{4}}^{4}\left(0,q\right)\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{L}$ : lattice, $q$ : nome and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators A&S Ref: 18.10.12, 18.10.13, 18.10.14, and 18.10.9, 18.10.10, 18.10.11 Referenced by: §23.22(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E4 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23

23.6.5	$\displaystyle\wp\left(z\right)-e_{1}$	$\displaystyle=\left(\frac{\pi\theta_{3}\left(0,q\right)\theta_{4}\left(0,q% \right)\theta_{2}\left(\pi z/(2\omega_{1}),q\right)}{2\omega_{1}\theta_{1}% \left(\pi z/(2\omega_{1}),q\right)}\right)^{2},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators A&S Ref: 18.10.5 Referenced by: §20.9(i), §23.22(i), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E5 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23
23.6.6	$\displaystyle\wp\left(z\right)-e_{2}$	$\displaystyle=\left(\frac{\pi\theta_{2}\left(0,q\right)\theta_{4}\left(0,q% \right)\theta_{3}\left(\pi z/(2\omega_{1}),q\right)}{2\omega_{1}\theta_{1}% \left(\pi z/(2\omega_{1}),q\right)}\right)^{2},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators A&S Ref: 18.10.5 Permalink: http://dlmf.nist.gov/23.6.E6 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23
23.6.7	$\displaystyle\wp\left(z\right)-e_{3}$	$\displaystyle=\left(\frac{\pi\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(\pi z/(2\omega_{1}),q\right)}{2\omega_{1}\theta_{1}% \left(\pi z/(2\omega_{1}),q\right)}\right)^{2}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators A&S Ref: 18.10.5 Referenced by: §20.9(i), §23.6(i), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E7 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23

23.6.8		$\eta_{1}=-\frac{\pi^{2}}{12\omega_{1}}\frac{\theta_{1}'''\left(0,q\right)}{% \theta_{1}'\left(0,q\right)}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $q$ : nome, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Referenced by: §23.22(i), §23.6(i) Permalink: http://dlmf.nist.gov/23.6.E8 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23

23.6.9		$\sigma\left(z\right)=2\omega_{1}\exp\left(\frac{\eta_{1}z^{2}}{2\omega_{1}}% \right)\frac{\theta_{1}\left(\pi z/(2\omega_{1}),q\right)}{\pi\theta_{1}'\left% (0,q\right)},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number A&S Ref: 18.10.8 Referenced by: §23.22(i), §23.6(i) Permalink: http://dlmf.nist.gov/23.6.E9 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23

23.6.10	$\displaystyle\sigma\left(\omega_{1}\right)$	$\displaystyle=2\omega_{1}\frac{\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\right)% \theta_{2}\left(0,q\right)}{\pi\theta_{1}'\left(0,q\right)},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathbb{L}$ : lattice, $q$ : nome, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Referenced by: §23.6(i) Permalink: http://dlmf.nist.gov/23.6.E10 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23
23.6.11	$\displaystyle\sigma\left(\omega_{2}\right)$	$\displaystyle=-2\omega_{1}\frac{\exp\left(\tfrac{1}{2}\eta_{1}(\omega_{1}\tau^% {2}+\omega_{3}-\omega_{2})\right)\theta_{3}\left(0,q\right)}{\pi q^{1/4}\theta% _{1}'\left(0,q\right)},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $q$ : nome, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $\tau$ : complex variable and $\eta_{j}$ : complex number Referenced by: Erratum (V1.1.1) for Equation (23.6.11) Permalink: http://dlmf.nist.gov/23.6.E11 Encodings: TeX, pMML, png Correction (effective with 1.1.1): The factor $2\omega_{1}i\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\tau^{2}\right)$ has been corrected to be $-2\omega_{1}\exp\left(\tfrac{1}{2}\eta_{1}(\omega_{1}\tau^{2}+\omega_{3}-% \omega_{2})\right)$ . See also: Annotations for §23.6(i), §23.6 and Ch.23
23.6.12	$\displaystyle\sigma\left(\omega_{3}\right)$	$\displaystyle=2i\omega_{1}\frac{\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\tau^{% 2}\right)\theta_{4}\left(0,q\right)}{\pi q^{1/4}\theta_{1}'\left(0,q\right)}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $q$ : nome, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $\tau$ : complex variable and $\eta_{j}$ : complex number Referenced by: §23.6(i), Erratum (V1.1.1) for Equation (23.6.12) Permalink: http://dlmf.nist.gov/23.6.E12 Encodings: TeX, pMML, png Correction (effective with 1.1.1): The factor $-2\omega_{1}\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\right)$ has been corrected to be $2i\omega_{1}\exp\left(\tfrac{1}{2}\eta_{1}\omega_{1}\tau^{2}\right)$ . See also: Annotations for §23.6(i), §23.6 and Ch.23

With $z=\ifrac{\pi u}{(2\omega_{1})}$ ,

23.6.13		$\zeta\left(u\right)=\frac{\eta_{1}}{\omega_{1}}u+\frac{\pi}{2\omega_{1}}\frac{% \mathrm{d}}{\mathrm{d}z}\ln\theta_{1}\left(z,q\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\ln\NVar{z}$ : principal branch of logarithm function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $\eta_{j}$ : complex number and $u$ : change of variable Referenced by: §23.22(i), §23.6(i) Permalink: http://dlmf.nist.gov/23.6.E13 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23

23.6.14		$\wp\left(u\right)=\left(\frac{\pi}{2\omega_{1}}\right)^{2}\left(\frac{\theta_{% 1}'''\left(0,q\right)}{3\theta_{1}'\left(0,q\right)}-\frac{{\mathrm{d}}^{2}}{{% \mathrm{d}z}^{2}}\ln\theta_{1}\left(z,q\right)\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\ln\NVar{z}$ : principal branch of logarithm function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $u$ : change of variable Referenced by: §23.6(i) Permalink: http://dlmf.nist.gov/23.6.E14 Encodings: TeX, pMML, png See also: Annotations for §23.6(i), §23.6 and Ch.23

23.6.15		$\frac{\sigma\left(u+\omega_{j}\right)}{\sigma\left(\omega_{j}\right)}=\exp% \left(\eta_{j}u+\frac{\eta_{1}u^{2}}{2\omega_{1}}\right)\frac{\theta_{j+1}% \left(z,q\right)}{\theta_{j+1}\left(0,q\right)},$
$j=1,2,3$ .
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\exp\NVar{z}$ : exponential function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $\eta_{j}$ : complex number and $u$ : change of variable Referenced by: §23.6(i), Erratum (V1.1.1) for Equation (23.6.15) Permalink: http://dlmf.nist.gov/23.6.E15 Encodings: TeX, pMML, png Correction (effective with 1.1.1): The factor $\exp\left(\eta_{j}u+\frac{\eta_{j}u^{2}}{2\omega_{1}}\right)$ has been corrected to be $\exp\left(\eta_{j}u+\frac{\eta_{1}u^{2}}{2\omega_{1}}\right)$ . Suggested 2021-02-10 by Jan Felipe van Diejen See also: Annotations for §23.6(i), §23.6 and Ch.23

For further results for the $\sigma$ -function see Lawden (1989, §6.2).

§23.6(ii) Jacobian Elliptic Functions

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Keywords:: Jacobian elliptic functions, Jacobian elliptic functions, Weierstrass elliptic functions, relations to other functions
Notes:: For (23.6.16) and (23.6.17) combine (23.6.2)–(23.6.4) with (22.2.2) and (20.7.5). For (23.6.18)–(23.6.20) combine (20.9.1) and (20.9.2) with (23.6.2)–(23.6.4). For (23.6.21)–(23.6.23) combine (23.6.5)–(23.6.7) with (22.2.4)–(22.2.9). For (23.6.24)–(23.6.26) combine (23.6.21)–(23.6.23) with §22.4(iii). (23.6.27)–(23.6.29) can be verified by matching periods, poles, and residues as in Lawden (1989, §8.11).
Referenced by:: §20.11(iii)
Permalink:: http://dlmf.nist.gov/23.6.ii
See also:: Annotations for §23.6 and Ch.23

Again, in Equations (23.6.16)–(23.6.26), $2\omega_{1},2\omega_{3}$ are any pair of generators of the lattice $\mathbb{L}$ and $e_{1},e_{2},e_{3}$ are given by (23.3.9).

23.6.16	$\displaystyle k^{2}$	$\displaystyle=\frac{e_{2}-e_{3}}{e_{1}-e_{3}},$
23.6.16	$\displaystyle{k^{\prime}}^{2}$	$\displaystyle=\frac{e_{1}-e_{2}}{e_{1}-e_{3}},$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 18.9.9 Referenced by: (23.22.1), item 1, §23.22(ii), §23.6(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.6(ii), §23.6 and Ch.23

23.6.17	$\displaystyle{K}^{2}$	$\displaystyle=(K\left(k\right))^{2}=\omega_{1}^{2}(e_{1}-e_{3}),$
23.6.17	$\displaystyle{{K^{\prime}}}^{2}$	$\displaystyle=(K\left(k^{\prime}\right))^{2}=\omega_{3}^{2}(e_{3}-e_{1}).$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 18.9.8 Referenced by: (23.22.1), §23.22(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E17 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.6(ii), §23.6 and Ch.23

23.6.18	$\displaystyle e_{1}$	$\displaystyle=\frac{{K}^{2}}{3\omega_{1}^{2}}(1+{k^{\prime}}^{2}),$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 18.9.1 Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E18 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23
23.6.19	$\displaystyle e_{2}$	$\displaystyle=\frac{{K}^{2}}{3\omega_{1}^{2}}(k^{2}-{k^{\prime}}^{2}),$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 18.9.2 Permalink: http://dlmf.nist.gov/23.6.E19 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23
23.6.20	$\displaystyle e_{3}$	$\displaystyle=-\frac{{K}^{2}}{3\omega_{1}^{2}}(1+k^{2}).$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus A&S Ref: 18.9.3 Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E20 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23

23.6.21	$\displaystyle\wp\left(z\right)-e_{1}$	$\displaystyle=\frac{{K}^{2}}{\omega_{1}^{2}}{\operatorname{cs}}^{2}\left(\frac% {K\!z}{\omega_{1}},k\right),$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E21 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23
23.6.22	$\displaystyle\wp\left(z\right)-e_{2}$	$\displaystyle=\frac{{K}^{2}}{\omega_{1}^{2}}{\operatorname{ds}}^{2}\left(\frac% {K\!z}{\omega_{1}},k\right),$
ⓘ Symbols: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus Permalink: http://dlmf.nist.gov/23.6.E22 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23
23.6.23	$\displaystyle\wp\left(z\right)-e_{3}$	$\displaystyle=\frac{{K}^{2}}{\omega_{1}^{2}}{\operatorname{ns}}^{2}\left(\frac% {K\!z}{\omega_{1}},k\right).$
ⓘ Symbols: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus A&S Ref: 18.9.11 Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E23 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23

23.6.24	$\displaystyle\wp\left(z+\omega_{1}\right)-e_{1}$	$\displaystyle=\left(\frac{Kk^{\prime}}{\omega_{1}}\right)^{2}{\operatorname{sc% }}^{2}\left(\frac{K\!z}{\omega_{1}},k\right),$
ⓘ Symbols: $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E24 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23
23.6.25	$\displaystyle\wp\left(z+\omega_{2}\right)-e_{2}$	$\displaystyle=-\left(\frac{Kkk^{\prime}}{\omega_{1}}\right)^{2}{\operatorname{% sd}}^{2}\left(\frac{K\!z}{\omega_{1}},k\right),$
ⓘ Symbols: $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/23.6.E25 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23
23.6.26	$\displaystyle\wp\left(z+\omega_{3}\right)-e_{3}$	$\displaystyle=\left(\frac{Kk}{\omega_{1}}\right)^{2}{\operatorname{sn}}^{2}% \left(\frac{K\!z}{\omega_{1}},k\right).$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus Referenced by: §23.6(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E26 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23

In (23.6.27)–(23.6.29) the modulus $k$ is given and $K=K\left(k\right)$ , ${K^{\prime}}=K\left(k^{\prime}\right)$ are the corresponding complete elliptic integrals (§19.2(ii)). Also, $\mathbb{L}_{\mspace{1.0mu}1}$ , $\mathbb{L}_{\mspace{1.0mu}2}$ , $\mathbb{L}_{\mspace{1.0mu}3}$ are the lattices with generators $(4K,2\mathrm{i}{K^{\prime}})$ , $(2K-2\mathrm{i}{K^{\prime}},2K+2\mathrm{i}{K^{\prime}})$ , $(2K,4\mathrm{i}{K^{\prime}})$ , respectively.

23.6.27		$\zeta\left(z\|\mathbb{L}_{\mspace{1.0mu}1}\right)-\zeta\left(z+2K\|\mathbb{L}_{% \mspace{1.0mu}1}\right)+\zeta\left(2K\|\mathbb{L}_{\mspace{1.0mu}1}\right)=% \operatorname{ns}\left(z,k\right),$
ⓘ Symbols: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex and $k$ : modulus Referenced by: §23.6(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E27 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23

23.6.28		$\zeta\left(z\|\mathbb{L}_{\mspace{1.0mu}2}\right)-\zeta\left(z+2K\|\mathbb{L}_{% \mspace{1.0mu}2}\right)+\zeta\left(2K\|\mathbb{L}_{\mspace{1.0mu}2}\right)=% \operatorname{ds}\left(z,k\right),$
ⓘ Symbols: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/23.6.E28 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23

23.6.29		$\zeta\left(z\|\mathbb{L}_{\mspace{1.0mu}3}\right)-\zeta\left(z+2\mathrm{i}{K^{% \prime}}\|\mathbb{L}_{\mspace{1.0mu}3}\right)-\zeta\left(2\mathrm{i}{K^{\prime}% }\|\mathbb{L}_{\mspace{1.0mu}3}\right)=\operatorname{cs}\left(z,k\right).$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $z$ : complex and $k$ : modulus Referenced by: §23.6(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/23.6.E29 Encodings: TeX, pMML, png See also: Annotations for §23.6(ii), §23.6 and Ch.23

Similar results for the other nine Jacobi functions can be constructed with the aid of the transformations given by Table 22.4.3.

For representations of the Jacobi functions $\operatorname{sn}$ , $\operatorname{cn}$ , and $\operatorname{dn}$ as quotients of $\sigma$ -functions see Lawden (1989, §§6.2, 6.3).

§23.6(iii) General Elliptic Functions

ⓘ

Keywords:: Weierstrass elliptic functions, elliptic functions, general, general elliptic functions, relations to other functions, representation as Weierstrass elliptic functions
Permalink:: http://dlmf.nist.gov/23.6.iii
See also:: Annotations for §23.6 and Ch.23

For representations of general elliptic functions (§23.2(iii)) in terms of $\sigma\left(z\right)$ and $\wp\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta\left(z\right)$ see Lawden (1989, §8.11).

§23.6(iv) Elliptic Integrals

ⓘ

Keywords:: Weierstrass elliptic functions, elliptic integrals, relations to other functions
Referenced by:: §19.10(i)
Permalink:: http://dlmf.nist.gov/23.6.iv
See also:: Annotations for §23.6 and Ch.23

Rectangular Lattice

ⓘ

See also:: Annotations for §23.6(iv), §23.6 and Ch.23

Let $z$ be on the perimeter of the rectangle with vertices $0,2\omega_{1},2\omega_{1}+2\omega_{3},2\omega_{3}$ . Then $t=\wp\left(z\right)$ is real (§§23.5(i)–23.5(ii)), and

23.6.30		$z=\frac{1}{2}\int_{t}^{\infty}\frac{\,\mathrm{d}u}{\sqrt{(u-e_{1})(u-e_{2})(u-% e_{3})}},$
$t\geq e_{1}$ , $z\in(0,\omega_{1}]$ ,
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $t$ Referenced by: §23.6(iv) Permalink: http://dlmf.nist.gov/23.6.E30 Encodings: TeX, pMML, png See also: Annotations for §23.6(iv), §23.6(iv), §23.6 and Ch.23

23.6.31	$\displaystyle z-\omega_{1}$	$\displaystyle=\frac{i}{2}\int_{t}^{e_{1}}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(% u-e_{2})(u-e_{3})}},$
$e_{2}\leq t\leq e_{1}$ , $z\in[\omega_{1},\omega_{1}+\omega_{3}]$ ,
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $t$ Permalink: http://dlmf.nist.gov/23.6.E31 Encodings: TeX, pMML, png See also: Annotations for §23.6(iv), §23.6(iv), §23.6 and Ch.23
23.6.32	$\displaystyle z-\omega_{3}$	$\displaystyle=\frac{1}{2}\int_{e_{3}}^{t}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(% e_{2}-u)(u-e_{3})}},$
$e_{3}\leq t\leq e_{2}$ , $z\in[\omega_{3},\omega_{1}+\omega_{3}]$ ,
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $t$ Permalink: http://dlmf.nist.gov/23.6.E32 Encodings: TeX, pMML, png See also: Annotations for §23.6(iv), §23.6(iv), §23.6 and Ch.23

23.6.33		$z=\frac{i}{2}\int_{-\infty}^{t}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(e_{2}-u)(e% _{3}-u)}},$
$t\leq e_{3}$ , $z\in(0,\omega_{3}]$ .
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $t$ Permalink: http://dlmf.nist.gov/23.6.E33 Encodings: TeX, pMML, png See also: Annotations for §23.6(iv), §23.6(iv), §23.6 and Ch.23

23.6.34		$2\omega_{1}=\int_{e_{1}}^{\infty}\frac{\,\mathrm{d}u}{\sqrt{(u-e_{1})(u-e_{2})% (u-e_{3})}}=\int_{e_{3}}^{e_{2}}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(e_{2}-u)(% u-e_{3})}},$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators Permalink: http://dlmf.nist.gov/23.6.E34 Encodings: TeX, pMML, png See also: Annotations for §23.6(iv), §23.6(iv), §23.6 and Ch.23

23.6.35		$2\omega_{3}=i\int_{e_{2}}^{e_{1}}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(u-e_{2})% (u-e_{3})}}=i\int_{-\infty}^{e_{3}}\frac{\,\mathrm{d}u}{\sqrt{(e_{1}-u)(e_{2}-% u)(e_{3}-u)}}.$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbb{L}$ : lattice and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators Referenced by: §23.6(iv) Permalink: http://dlmf.nist.gov/23.6.E35 Encodings: TeX, pMML, png See also: Annotations for §23.6(iv), §23.6(iv), §23.6 and Ch.23

For (23.6.30)–(23.6.35) and further identities see Lawden (1989, §6.12).

See also §§19.2(i), 19.14, and Erdélyi et al. (1953b, §13.14).

For relations to symmetric elliptic integrals see §19.25(vi).

General Lattice

ⓘ

Keywords:: Weierstrass elliptic functions, elliptic integrals, relations to other functions
See also:: Annotations for §23.6(iv), §23.6 and Ch.23

Let $z$ be a point of $\mathbb{C}$ different from $e_{1},e_{2},e_{3}$ , and define $w$ by

23.6.36		$w=\int_{z}^{\infty}\frac{\,\mathrm{d}u}{\sqrt{4u^{3}-g_{2}u-g_{3}}}=\frac{1}{2% }\int_{z}^{\infty}\frac{\,\mathrm{d}u}{\sqrt{(u-e_{1})(u-e_{2})(u-e_{3})}},$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex Referenced by: (19.25.35), §19.25(vi) Permalink: http://dlmf.nist.gov/23.6.E36 Encodings: TeX, pMML, png See also: Annotations for §23.6(iv), §23.6(iv), §23.6 and Ch.23

where the integral is taken along any path from $z$ to $\infty$ that does not pass through any of $e_{1},e_{2},e_{3}$ . Then $z=\wp\left(w\right)$ , where the value of $w$ depends on the choice of path and determination of the square root; see McKean and Moll (1999, pp. 87–88 and §2.5).

23.5 Special Lattices 23.7 Quarter Periods

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