Page 193 -

170 9. Elliptic and Hypergeometric Functions

(2.4) Corollary. An elliptic function cannot have only one simple pole mod L.

For a meromorphic function f (z) we denote m = ord a f (z) when f (z) =
m
(z − a) g(z), where g(z) is holomorphic near a and g(a) = 0. When ord a f (z)> 0
the point a is a zero of f and when ord a f (z)< 0, it is a pole of f .
(2.5) Theorem. Let f (z) be an elliptic function with no zeros or poles on the bound-
ary of the fundamental parallelogram P. Then

ord a f (z) = 0,
a∈P
and

a · ord a f (z) ≡ 0 (mod L).
a∈P
Proof. The zeros and poles of f are the simple poles of f /f , and the multiplicities

are the residues of f /f counted positive for zeros and negative for poles. The first

relation follows from (2.3) applied to the elliptic function f /f which is an elliptic
function.
The second relation follows by considering the integral
1 ( f (z) f (z) ord a f

z dz, where = + holomorphic function.
2πi ∂ P f (z) f (z) z − a

By the residue calculus this integral equals the sum a · ord a ( f ). Consider the
a∈P
part of P from z 0 to z 0 + ω 1 and from z 0 + ω 2 to z 0 + ω 1 + ω 2 , and calculate this
part of the integral around a

1 ( z 0 +ω 1 ( z 0 +ω 1 +ω 2 f (z) ω 2 ( a+ω 1 f (z)
− z dz = dz.
2πi f (z) 2πi a f (z)
z 0 z 0 +ω 2
Except for the factor ω 2 the right-hand expression is the winding number around 0
of the closed curve parametrized by f (a + tω 1 ) for 0 ≤ t ≤ 1. Hence this part of
the integral is in Zω 2 and by a similar argument the other part is in Zω 1 . Hence the
above integral around ∂ P representing the sum of a · ord a ( f ) over P is in L. This
proves the theorem.

(2.6) Remarks. The set of all elliptic functions M L associated with the lattice L
forms a field under the usual operations of addition and multiplication of functions.
The field of complex numbers is always the subfield of the constant elliptic functions
M L , and at this point these are the only examples of elliptic functions. In view of the
previous two theorems the simplest nonconstant elliptic functions would have either
a single double pole with residue zero in 1/2L or two simple poles whose sum is
in the lattice with two residues adding to zero. There are constructions for elliptic
functions having such simple singularities which are due to Weierstrass and Jacobi,
respectively.