Genus‑1 canonical factors are multipliable under the standard hypothesis ∑ 1/‖z i‖² < ∞.
This isolates the analytic input used repeatedly when building canonical products:
for fixed s, the family i ↦ weierstrassE 1 (s / z i) is an infinite product that converges.
General-rank versions #
theorem
Hadamard.OrderOne.summable_norm_weierstrass_E_sub_one_of_summable_inv_norm_pow
{ι : Type u_1}
{z : ι → ℂ}
{p : ℕ}
(hz0 : ∀ (i : ι), z i ≠ 0)
(h : Summable fun (i : ι) => 1 / ‖z i‖ ^ (p + 1))
(s : ℂ)
:
Rank-p analogue of summable_norm_weierstrass_E_one_sub_one_of_summable_inv_norm_sq.
Under ∑ 1/‖z i‖^(p+1) < ∞, the family ‖E_p(s/z i) - 1‖ is summable for every s : ℂ.
theorem
Hadamard.OrderOne.multipliable_weierstrass_E_of_summable_inv_norm_pow
{ι : Type u_1}
{z : ι → ℂ}
{p : ℕ}
(hz0 : ∀ (i : ι), z i ≠ 0)
(h : Summable fun (i : ι) => 1 / ‖z i‖ ^ (p + 1))
(s : ℂ)
:
Multipliable fun (i : ι) => weierstrassE p (s / z i)
Rank-p analogue of multipliable_weierstrass_E_one_of_summable_inv_norm_sq.
theorem
Hadamard.OrderOne.multipliable_weierstrass_E_one_of_summable_inv_norm_sq
{ι : Type}
{z : ι → ℂ}
(hz0 : ∀ (i : ι), z i ≠ 0)
(h : Summable fun (i : ι) => 1 / ‖z i‖ ^ 2)
(s : ℂ)
:
Multipliable fun (i : ι) => weierstrassE 1 (s / z i)