58.3. EXERCISES 1851

58.3 Exercises1. Show the set of modular transformations is a group. Also show those modular trans-

formations which are congruent mod2 to the identity as described above is a sub-group.

2. Suppose f is an elliptic function with period module M. If {w1,w2} and {w′1,w′2}are two bases, show that the resulting period parallelograms resulting from the twobases have the same area.

3. Given a module of periods with basis {w1,w2} and letting a typical element of thismodule be denoted by w as described above, consider the product

σ (z)≡ z ∏w̸=0

(1− z

w

)e(z/w)+ 1

2 (z/w)2.

Show this product converges uniformly on compact sets, is an entire function, andsatisfies

σ′ (z)/σ (z) = ζ (z)

where ζ (z) was defined above as a primitive of ℘(z) and is given by

ζ (z) =1z+ ∑

w ̸=0

1z−w

+z

w2 +1w.

4. Show ζ (z+wi) = ζ (z)+η i where η i is a constant.

5. Let Pa be the parallelogram shown in the following picture.

0w1

w2

a

Show that 12πi∫

∂Paζ (z)dz = 1 where the contour is taken once around the parallelo-

gram in the counter clockwise direction. Next evaluate this contour integral directlyto obtain Legendre’s relation,

η1w2−η2w1 = 2πi.

6. For σ defined in Problem 3, 4 explain the following steps. For j = 1,2

σ ′ (z+w j)

σ (z+w j)= ζ (z+w j) = ζ (z)+η j =

σ ′ (z)σ (z)

+η j