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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.9 Laurent and Other Power Series

Let z0(0) be the nearest lattice point to the origin, and define

23.9.1 cn=(2n-1)w𝕃\{0}w-2n,
n=2,3,4,.

Then

23.9.2 (z)=1z2+n=2cnz2n-2,
0<|z|<|z0|,
23.9.3 ζ(z)=1z-n=2cn2n-1z2n-1,
0<|z|<|z0|.

Here

23.9.4 c2 =120g2,
c3 =128g3,
23.9.5 cn=3(2n+1)(n-3)m=2n-2cmcn-m,
n4.

Explicit coefficients cn in terms of c2 and c3 are given up to c19 in Abramowitz and Stegun (1964, p. 636).

For j=1,2,3, and with ej as in §23.3(i),

23.9.6 (ωj+t)=ej+(3ej2-5c2)t2+(10c2ej+21c3)t4+(7c2ej2+21c3ej+5c22)t6+O(t8),

as t0. 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/(z)0.

For z

23.9.7 σ(z)=m,n=0am,n(10c2)m(56c3)nz4m+6n+1(4m+6n+1)!,

where a0,0=1, am,n=0 if either m or n<0, and

23.9.8 am,n=3(m+1)am+1,n-1+163(n+1)am-2,n+1-13(2m+3n-1)(4m+6n-1)am-1,n.

For am,n with m=0,1,,12 and n=0,1,,8, see Abramowitz and Stegun (1964, p. 637).