§23.9 Laurent and Other 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).
Keywords:: Laurent series, Weierstrass elliptic functions, power series
Referenced by:: §20.6, §20.6
Permalink:: http://dlmf.nist.gov/23.9

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, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\setminus$ : 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

Then

23.9.2 $\mathop{\wp\/}\nolimits\!\left(z\right)=\frac{1}{z^{2}}+\sum _{{n=2}}^{\infty}c_{n}z^{{2n-2}},$ $0<|z|<|z_{0}|$ ,

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, $n$ : 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

23.9.3 $\mathop{\zeta\/}\nolimits\!\left(z\right)=\frac{1}{z}-\sum _{{n=2}}^{\infty}\frac{c_{n}}{2n-1}z^{{2n-1}},$ $0<|z|<|z_{0}|$ .

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, $n$ : 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

Here

23.9.4

$c_{2}=\frac{1}{20}g_{2},$

$c_{3}=\frac{1}{28}g_{3},$

Symbols:: $g_{2}$ , $g_{3}$ : lattice invariants 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

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

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 $\mathop{\wp\/}\nolimits\!\left(\omega _{j}+t\right)=e_{j}+(3e_{j}^{2}-5c_{2})t^{2}+(10c_{2}e_{j}+21c_{3})t^{4}+(7c_{2}e_{j}^{2}+21c_{3}e_{j}+5c_{2}^{2})t^{6}+\mathop{O\/}\nolimits\!\left(t^{8}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $c_{n}$ and $t$ : variable
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E6
Encodings:: TeX, pMML, png

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 $1/\mathop{\wp\/}\nolimits\!\left(z\right)\to 0$ .