# §23.9 Laurent and Other Power Series

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

 23.9.1 $c_{n}=(2n-1)\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-2n},$ $n=2,3,4,\dots$.

Then

 23.9.2 $\mathop{\wp\/}\nolimits\!\left(z\right)=\frac{1}{z^{2}}+\sum_{n=2}^{\infty}c_{% n}z^{2n-2},$ $0<|z|<|z_{0}|$,
 23.9.3 $\mathop{\zeta\/}\nolimits\!\left(z\right)=\frac{1}{z}-\sum_{n=2}^{\infty}\frac% {c_{n}}{2n-1}z^{2n-1},$ $0<|z|<|z_{0}|$.

Here

 23.9.4 $\displaystyle c_{2}$ $\displaystyle=\frac{1}{20}g_{2},$ $\displaystyle c_{3}$ $\displaystyle=\frac{1}{28}g_{3},$ 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 $\mathop{\wp\/}\nolimits\!\left(\omega_{j}+t\right)=e_{j}+(3e_{j}^{2}-5c_{2})t^% {2}+(10c_{2}e_{j}+21c_{3})t^{4}+(7c_{2}e_{j}^{2}+21c_{3}e_{j}+5c_{2}^{2})t^{6}% +\mathop{O\/}\nolimits\!\left(t^{8}\right),$

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

For $z\in\Complex$

 23.9.7 $\mathop{\sigma\/}\nolimits\!\left(z\right)=\sum_{m,n=0}^{\infty}a_{m,n}(10c_{2% })^{m}(56c_{3})^{n}\frac{z^{4m+6n+1}}{(4m+6n+1)!},$

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

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