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172 9. Elliptic and Hypergeometric Functions

1 ) 1 1 *
℘(z; L) = ℘(z) = + − ,
z 2 (z − ω) 2 ω 2
ω∈L−0
1 ) 1 1 z *
ζ(z; L) = ζ(z) = + + + 2 .
z z − ω ω ω
ω∈L−0
Since
1 1 2ωz − z 2
− =
2
(z − ω) 2 ω 2 ω (z − ω) 2
and

1 1 z z 2
+ + =
2
z − ω ω ω 2 ω (z − ω)
3
both expressions behave like 1/ω for |ω| large. Thus by Lemma (3.1) the above
infinite sums converge uniformly on compact subsets of C − L and define meromor-
phic functions on C with, respectively, a simple double pole and a simple first-order

pole at each ω ∈ L. Observe that the derivative ζ (z) =−℘(z), and we have the
following expression
1

℘ (z; L) = ℘ (z) =−2
(z − ω) 3
ω∈L−0

for the derivative of the ℘-function. It is clear from the formula that ℘ (z) is an
elliptic function with a third-order pole at each ω ∈ L having zero residue. From
the formula for ℘(z) it follows that ℘(z) is an even function, i.e., ℘(z) = ℘(−z),

and ℘ (z) is clearly an odd function. Since also ℘ (z + ω i ) = ℘ (z) we deduce that
℘(z+ω i ) = ℘(z)+c i for i = 1, 2, where c i are constant, and by setting z =−ω i /2,
we calculate

ω i −ω i −ω i ω i

℘ = ℘ + ω i = ℘ + c i = ℘ + c i .
2 2 2 2
Thus each c i = 0and ℘(z) is an elliptic function. The Weierstrass zeta function,
which should not be confused with the Riemann zeta function, is not an elliptic
function by (2.4). See the exercises for further properties.

With the elliptic functions ℘(z) and ℘ (z) we obtain all others in the sense made
precise in the next theorem.
(3.3) Theorem. For a lattice L the field M L of elliptic functions is

C(℘(z, L), ℘ (z, L)),

the field generated by ℘ and ℘ over the constants, and the field of even elliptic
function is C(℘(z, L)).