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§5. Preliminaries on Hypergeometric Functions 179

Exercises

1. Prove the following determinantal relation

℘(z 1 ) ℘ (z 1 )
1

℘(z 2 ) ℘ (z 2 ) 1 = 0.

℘(z 1 + z 2 ) −℘ (z 1 + z 2 ) 1

2. For σ(z) as introduced in Exercise 3, §3, show that
σ(z 1 − z 2 )σ(z 1 + z 2 )
℘(z 1 ) − ℘(z 2 ) =− 2 2
σ(z 1 ) σ(z 2 )
and
σ(2z)

℘ (x) =− 4 .
σ(z)
3. Next derive the relation

℘ (z 1 )
= ζ(z 1 − z 2 ) + ζ(z 1 + z 2 ) − 2ζ(z 1 ).
℘(z 1 ) − ℘(z 2 )
4. Derive the addition formula

1 ℘ (z 1 ) − ℘ (z 2 )
ζ(z 1 + z 2 ) = ζ(z 1 ) + ζ(z 2 ) + .
2 ℘(z 1 ) − ℘(z 2 )
From this formula derive the addition formulas in (4.4).
§5. Preliminaries on Hypergeometric Functions

In order to find the complex torus associated with an elliptic curve, we will use some
special functions which we introduce in this section.
(5.1) Definition. The gamma function
(s) is defined for Re(s)> 0bythe follow-
ing integral:
∞ dx
(
s −x

(s) = x e .
0 x
Due to the exponential factor e −x the integral at ∞ converges for all s and for
Re(s)> 0 the factor x s−1 is integrable near 0.
(5.2) Remark. For Re(s)> 1 an easy integration by parts, which we leave as an
exercise, gives the relation

(s) = (s − 1)
(s − 1).
∞ −x

Since
(1) = e dx = 1, we prove inductively that for a natural number n we
0
have