An Elliptic Function – The Weierstrass Function

Definition W.1 An elliptic function f (z) is a non constant meromorphic function on C that is doubly periodic. That is, there are two nonzero complex numbers ω 1 , ω 2 whose ratio is not real, such that f (z + ω 1) = f (z) and f (z + ω 2) = f (z). Fix two real numbers β, γ > 0. The Weierstrass function with primitive periods γ and iβ is the function ℘ : C → C defined by ℘(z) = 1 z 2 + ω∈γZ Z⊕iβZ Z ω =0 1 (z − ω) 2 − 1 ω 2 It is an important example of an elliptic function. Its elementary properties are given in Problem W.1 Prove that a) For each fixed z ∈ C\(γZ Z⊕iβZ Z), the series ω∈γZ Z⊕iβZ Z ω =0 1 (z−ω) 2 − 1 ω 2 converges absolutely. The convergence is uniform on compact subsets of C \ (γZ Z ⊕ iβZ Z). b) ℘(z) is analytic on C \ (γZ Z ⊕ iβZ Z). c) ℘(z + ζ) = ℘(z) for all ζ ∈ γZ Z ⊕ iβZ Z. d) ℘(−z) = ℘(z). e) ℘(z) = ℘(¯ z) for all C \ (γZ Z ⊕ iβZ Z). f) ℘(x) and ℘(x + i β 2) are real for all x ∈ IR and ℘(iy) and ℘(iy + γ 2) are real for all y ∈ IR. The following Lemma is one of the main properties of elliptic functions. It proves that an elliptic function takes each value the same number of times and that number is just the sum of the degrees of its poles. Theorem W.2 Let f (z) be an elliptic function with periods ω 1 , ω 2. Set Ω = ω 1 Z Z + ω 2 Z Z. Suppose that f (z) has poles of order n 1 , · · · , n k at p 1 +Ω, · · · , p k +Ω and is analytic elsewhere. Let c be any complex number. Suppose that f (z) − c has zeroes of order m 1 , · · · , m h at (1)