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12 Weierstraß Functions
12.2 Fundamental Equations and Computation
The Weierstraß elliptic ℘ function is a double periodic function on a complex lattice.
The lattice generators are 2ω 1 and 2ω 3 with
ω 3
ω 1
) > 0, and ω 2 = −ω 1 −ω 3 , and the
lattice parameter τ =
ω 3
ω 1
with nome q = exp(iπτ ), see Eq. (9.2). The Weierstraß
elliptic function is defined as
℘ (z) =
1
z 2 +
(m,n) =(0,0)
1
(z − mω 1 − nω 3 ) 2 −
1
(mω 1 + nω 3 ) 2 ,
(12.1)
and thus by construction, this function is double periodic with fundamental periods
equal 2ω 1 , 2ω 3 , and an even function ℘ (−z) = ℘ (z).
The Weierstraß ζ function is defined as
dζ(z)
dz
= −℘ (z),
(12.2a)
and therefore
ζ(z) =
1
z
+
(m,n) =(0,0)
1
z − mω 1 − nω 3 )
+
1
mω 1 + nω 3
+
z
(mω 1 + nω 3 ) 2 .
(12.2b)
The Weierstraß ζ function is an odd function ζ(−z) = −ζ(z) and quasiperiodic
ζ(z + 2ω i ) = ζ(z) + 2η i with η i = ζ(ω i ). Finally, the Weierstraß σ function is
defined as
1
σ (z)
dσ (z)
dz
= ζ(z),
(12.3a)
or alternatively
σ (z)=
(m,n) =(0,0)
1 −
z
mω 1 + nω 3 )
exp
z
mω 1 + nω 3
+
z 2
2(mω 1 + nω 3 ) 2
.
(12.3b)
The σ -function is a quasiperiodic odd function σ (−z) = −σ (z).
The complex numbers 0, 2ω 1 , 2ω 2 , and 2ω 3 define a parallelogram on the
complex plane and the set of points 2mω 1 + 2nω 3 with (m, n) ∈ Z a lattice. The