1. Weierstrass ζ-function
Let ω1 , ω2 be two complex numbers such that Im τ > 0 where τ = ω2 /ω1 . Let L =
Zω1 ⊕ Zω2 be the lattice generated by {ω1 , ω2 }. The Weierstrass ζ function is defined by
X 1
1
1
z2
ζ(z) = +
+ +
z
z − ω ω ω2
ω∈L\{0}
It follows from the definition of ζ and ℘(z) that
ζ 0 (z) = −℘(z).
Since ℘(z) is an even function, ζ is an odd function, i.e. ζ(−z) = −ζ(z). If we write
∞
℘(z) =
X
1
+
bn z 2n
2
z
n=1
in a neighborhood of 0, then
∞
ζ(z) =
1 X bn
−
z 2n+1
z
2n + 1
n=1
in a neighborhood of 0. By integrating the relation ℘(z + ω1 ) = ℘(ω), we obtain ζ(z + ω1 ) =
ζ(z)+C, where C is a constant to be determined. Substituting z = −ω1 /2 into this equation
and using ζ(−z) = −ζ(z), we find C = 2ζ(ω1 /2). Hence we obtain ζ(z+ω1 ) = ζ(z)+2ζ ω21 .
In fact, we have
ω i
ζ(z + ωi ) = ζ(z) + 2ζ
, i = 1, 2, 3,
2
where ω3 = ω1 + ω2 . We denote
ω i
, i = 1, 2, 3.
ηi = ζ
2
Theorem 1.1. The Weierstrass ζ-function is an odd function of z such that ζ 0 (z) = −℘(z)
and ζ(z + ωi ) = ζ(z) + 2ηi for i = 1, 2, 3. It is holomorphic on C \ Λ and has pole at ω ∈ Λ
of residue 1.
Theorem 1.2. (Legendre relation)
Let us for simplicity denote
Q
η1 ω2 − η2 ω1 = πi.
Q0
P
P0
ω∈Λ\{0} by
ω and
ω∈Λ\{0} by
ω . Let us define
z2
E(z) = (1 − z)ez+ 2 ,
z ∈ C.
Lemma 1.1. For |z| ≤ 1/2,
|E(z) − 1| ≤ 2|z|3 .
This lemma implies that the finite product
Y0 z E
ω
ω
converges absolutely and uniformly on |z| ≤ R for every R > 0. This allows us to define an
entire function, called the Weierstrass σ-function by
Y0 z Y0 z z + z2
σ(z) = z
E
=z
1−
e ω 2ω .
ω
ω
ω
ω
1
2
After taking the principal branch of log, it follows from the definition that
d
log σ(z) = ζ(z).
dz
The σ function is an odd function of z. Integrating ζ(z + ωi ) = ζ(z) + 2ηi , we obtain
σ(z + ωi ) = ci σ(z)e2ηi z
where ci are constant to be determined. Substituting z = −ωi /2 into the equation and
using σ being odd, we find ci = −e−2ηi ωi . In other words, we obtain
ωi
(1.1)
σ(z + ω ) = −σ(z)e2ηi (z+ 2 )
i
Theorem 1.3. Let f (z) be an elliptic function of period {ω1 , ω2 }. Let a1 , · · · , an be the
zeros and b1 , · · · , bn be the poles of f (z) repeated according to their multiplicity which lie
in a fundamental period parallelogram P so that
n
n
X
X
ai ≡
bj mod L.
i=1
j=1
Then there exists a constant c ∈ C such that for all z ∈ C,
σ(z − a1 ) · · · σ(z − an )
.
(1.2)
f (z) = c ·
σ(z − b1 ) · · · σ(z − bn )
Proof. Let g(z) be the meromorphic function defined by the right hand side of equation
(1.2). Equation (1.1) implies
n
(−1)n e2η1 (nz+ 2 ω1 −a1 −···−an ) σ(z − a1 ) · · · σ(z − an )
g(z + ω1 ) =
n
(−1)n e2η1 (nz+ 2 ω1 −b1 −···−bn ) σ(z − b ) · · · σ(z − b )
1
n
= g(z).
Similarly, we have g(z + τ ) = g(z). This shows that g(z) is an elliptic function of periods
{1, τ }. The quotient f (z)/g(z) is again elliptic without poles. By Lemma ??, f (z)/g(z) is
a constant function. This completes the proof of our assertion.