23 Weierstrass Elliptic and Modular Functions Weierstrass Elliptic Functions 23.8 Trigonometric Series and Products 23.10 Addition Theorems and Other Identities

§23.9 Laurent and Other Power Series

ⓘ

Keywords:: Laurent series, Weierstrass elliptic functions, power series
Notes:: For (23.9.2)–(23.9.5) equate coefficients in (23.2.4), (23.2.5), and also apply (23.10.1), (23.10.2). The first two coefficients in the Maclaurin expansion (23.9.6) are given by (23.3.9), (23.2.10); the others are obtained from §23.3(ii) combined with (23.9.4). (23.9.7) follows from (23.2.8) and (23.9.3).
Referenced by:: §20.6, §20.6
Permalink:: http://dlmf.nist.gov/23.9
See also:: Annotations for Ch.23

Let $z_{0}(\neq 0)$ be the nearest lattice point to the origin, and define

23.9.1		$c_{n}=(2n-1)\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-2n},$
$n=2,3,4,\dots$ .
ⓘ Symbols: $\in$ : element of, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice, $n$ : integer and $c_{n}$ Referenced by: §20.6 Permalink: http://dlmf.nist.gov/23.9.E1 Encodings: TeX, pMML, png See also: Annotations for §23.9 and Ch.23

Then

23.9.2		$\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{n=2}^{\infty}c_{n}z^{2n-2},$
$0<\|z\|<\|z_{0}\|$ ,
ⓘ 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 $c_{n}$ A&S Ref: 18.5.1 Referenced by: §23.9, §23.9 Permalink: http://dlmf.nist.gov/23.9.E2 Encodings: TeX, pMML, png See also: Annotations for §23.9 and Ch.23

23.9.3		$\zeta\left(z\right)=\frac{1}{z}-\sum_{n=2}^{\infty}\frac{c_{n}}{2n-1}z^{2n-1},$
$0<\|z\|<\|z_{0}\|$ .
ⓘ 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, $n$ : integer, $z$ : complex and $c_{n}$ A&S Ref: 18.5.5 Referenced by: §23.9 Permalink: http://dlmf.nist.gov/23.9.E3 Encodings: TeX, pMML, png See also: Annotations for §23.9 and Ch.23

Here

23.9.4	$\displaystyle c_{2}$	$\displaystyle=\frac{1}{20}g_{2},$
23.9.4	$\displaystyle c_{3}$	$\displaystyle=\frac{1}{28}g_{3},$
ⓘ Symbols: $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice and $c_{n}$ A&S Ref: 18.5.2 Referenced by: §23.9 Permalink: http://dlmf.nist.gov/23.9.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.9 and Ch.23

23.9.5		$c_{n}=\frac{3}{(2n+1)(n-3)}\sum_{m=2}^{n-2}c_{m}c_{n-m},$
$n\geq 4$ .
ⓘ Symbols: $n$ : integer, $m$ : integer and $c_{n}$ A&S Ref: 18.5.3 Referenced by: §23.9 Permalink: http://dlmf.nist.gov/23.9.E5 Encodings: TeX, pMML, png See also: Annotations for §23.9 and Ch.23

Explicit coefficients $c_{n}$ in terms of $c_{2}$ and $c_{3}$ are given up to $c_{19}$ in Abramowitz and Stegun (1964, p. 636).

For $j=1,2,3$ , and with $e_{j}$ as in §23.3(i),

23.9.6		$\wp\left(\omega_{j}+t\right)=e_{j}+(3e_{j}^{2}-5c_{2})t^{2}+(10c_{2}e_{j}+21c_% {3})t^{4}+(7c_{2}e_{j}^{2}+21c_{3}e_{j}+5c_{2}^{2})t^{6}+O\left(t^{8}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $c_{n}$ and $t$ : variable Referenced by: §23.9 Permalink: http://dlmf.nist.gov/23.9.E6 Encodings: TeX, pMML, png See also: Annotations for §23.9 and Ch.23

as $t\to 0$ . For the next four terms see Abramowitz and Stegun (1964, (18.5.56)). Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as $1/\wp\left(z\right)\to 0$ .

For $z\in\mathbb{C}$

23.9.7		$\sigma\left(z\right)=\sum_{m,n=0}^{\infty}a_{m,n}(10c_{2})^{m}(56c_{3})^{n}% \frac{z^{4m+6n+1}}{(4m+6n+1)!},$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $!$ : factorial (as in $n!$ ), $\mathbb{L}$ : lattice, $n$ : integer, $m$ : integer, $z$ : complex, $c_{n}$ and $a_{m,n}$ : coefficients A&S Ref: 18.5.6 Referenced by: §23.9 Permalink: http://dlmf.nist.gov/23.9.E7 Encodings: TeX, pMML, png See also: Annotations for §23.9 and Ch.23

where $a_{0,0}=1$ , $a_{m,n}=0$ if either $m$ or $n<0$ , and

23.9.8		$a_{m,n}=3(m+1)a_{m+1,n-1}+\tfrac{16}{3}(n+1)a_{m-2,n+1}-\tfrac{1}{3}(2m+3n-1)(% 4m+6n-1)a_{m-1,n}.$
ⓘ Symbols: $n$ : integer, $m$ : integer and $a_{m,n}$ : coefficients A&S Ref: 18.5.8 Permalink: http://dlmf.nist.gov/23.9.E8 Encodings: TeX, pMML, png See also: Annotations for §23.9 and Ch.23

For $a_{m,n}$ with $m=0,1,\dots,12$ and $n=0,1,\dots,8$ , see Abramowitz and Stegun (1964, p. 637).

23.8 Trigonometric Series and Products 23.10 Addition Theorems and Other Identities

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§23.9 Laurent and Other Power Series

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