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DLMF: §23.14 Integrals ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

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23 Weierstrass Elliptic and Modular Functions Weierstrass Elliptic Functions 23.13 Zeros 23.15 Definitions

§23.14 Integrals

ⓘ

Keywords:: Weierstrass elliptic functions, integrals
Notes:: To verify these results differentiate and use (23.2.7), §23.3(ii).
Permalink:: http://dlmf.nist.gov/23.14
See also:: Annotations for Ch.23

23.14.1		$\int\wp\left(z\right)\,\mathrm{d}z=-\zeta\left(z\right),$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex A&S Ref: 18.6.1 Permalink: http://dlmf.nist.gov/23.14.E1 Encodings: TeX, pMML, png See also: Annotations for §23.14 and Ch.23

23.14.2		$\int{\wp}^{2}\left(z\right)\,\mathrm{d}z=\frac{1}{6}\wp'\left(z\right)+\frac{1% }{12}g_{2}z,$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex A&S Ref: 18.7.1 Permalink: http://dlmf.nist.gov/23.14.E2 Encodings: TeX, pMML, png See also: Annotations for §23.14 and Ch.23

23.14.3		$\int{\wp}^{3}\left(z\right)\,\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-% \frac{3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex A&S Ref: 18.7.2 Permalink: http://dlmf.nist.gov/23.14.E3 Encodings: TeX, pMML, png See also: Annotations for §23.14 and Ch.23

For further integrals see Gröbner and Hofreiter (1949, Vol. 1, pp. 161–162), Gradshteyn and Ryzhik (2015, §5.14), and Prudnikov et al. (1990, pp. 51–52).

23.13 Zeros 23.15 Definitions

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