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174 9. Elliptic and Hypergeometric Functions

Exercises

1. Show that ζ(z + ω) = ζ(z) + η(ω), where η(ω) are constants.
2. For η(ω i ) = η i derive Legendre’s relation

η 1 ω 2 − η 2 ω 1 = 2πi

by considering ζ(z)dz, where P is a fundamental parallelogram whose boundary is
∂ P
disjoint from L.
3. Show that the following infinite product converges and represents an entire function:
z
z ) z 1 2 *
σ(z) = z 1 − · exp + .
ω ω 2 ω
ω∈L−0
Show also that ζ(z) = σ (z)/σ(z) = (d/dz) log σ(z).

4. For L = Zω 1 + Zω 2 and ζ(z + ω i ) = ζ(z) + η i show that
" #
ω i
σ (z + ω i ) =−σ(z)exp η i z + .
2
5. Show that any elliptic function can be written as
σ(z − a i )
n
c .
σ(z − b i )
i=1
where c is a constant.

§4. The Differential Equation for ℘(z)

2

Since ℘ (z) is an odd elliptic function, its square ℘ (z) is an even elliptic function,
and by (3.3) it is a rational function in ℘(z). In this section we will prove that this
rational function is a cubic polynomial. To do this, we consider the Laurent develop-
ment of ζ(z, L), ℘(z, L),and ℘ (z, L) at the origin.

From the geometric series we have
1 1 n
z
=−
z − ω ω ω
0≤n
and
1 1 z 1 n
z
+ + =−
z − ω ω ω 2 ω ω
2≤n
which converge for |z|≤|ω|. Thus
) *
1 1 1 z
ζ(z, L) = + + +
z z − ω ω ω 2
ω∈L−0
1 z n
= −
z ω n+1
ω∈L−0,2≤n