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DLMF: §23.7 Quarter Periods ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

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23 Weierstrass Elliptic and Modular Functions Weierstrass Elliptic Functions 23.6 Relations to Other Functions 23.8 Trigonometric Series and Products

§23.7 Quarter Periods

ⓘ

Keywords:: Weierstrass elliptic functions, quarter periods
Notes:: See Lawden (1989, p. 182).
Permalink:: http://dlmf.nist.gov/23.7
See also:: Annotations for Ch.23

23.7.1		$\displaystyle\wp\left(\tfrac{1}{2}\omega_{1}\right)$	$\displaystyle=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}=e_{1}+\omega_{1}^{-2}(K% \left(k\right))^{2}k^{\prime},$
ⓘ 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, $K\left(\NVar{k}\right)$ : 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: 18.3.18 Permalink: http://dlmf.nist.gov/23.7.E1 Encodings: TeX, pMML, png See also: Annotations for §23.7 and Ch.23
23.7.2		$\displaystyle\wp\left(\tfrac{1}{2}\omega_{2}\right)$	$\displaystyle=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})}=e_{2}-i\omega_{1}^{-2}(% K\left(k\right))^{2}kk^{\prime},$
ⓘ 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, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 18.3.18 Permalink: http://dlmf.nist.gov/23.7.E2 Encodings: TeX, pMML, png See also: Annotations for §23.7 and Ch.23
23.7.3		$\displaystyle\wp\left(\tfrac{1}{2}\omega_{3}\right)$	$\displaystyle=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}=e_{3}-\omega_{1}^{-2}(K% \left(k\right))^{2}k,$
ⓘ 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, $K\left(\NVar{k}\right)$ : 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: 18.3.18 Permalink: http://dlmf.nist.gov/23.7.E3 Encodings: TeX, pMML, png See also: Annotations for §23.7 and Ch.23

where $k,k^{\prime}$ and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.

23.6 Relations to Other Functions 23.8 Trigonometric Series and Products

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