apply DifferentiableOn.div <| DifferentiableOn.cexp <| DifferentiableOn.const_mul differentiableOn_id _
x σ σ' σ'' T : ℝ
xpos : 0 < x
⊢ HolomorphicOn (fun s => cexp ((↑x).log * s) / (s * (s + 1))) {0, -1}ᶜ
case hd
x σ σ' σ'' T : ℝ
xpos : 0 < x
⊢ DifferentiableOn ℂ (fun x => x * (x + 1)) {0, -1}ᶜ

case hx
x σ σ' σ'' T : ℝ
xpos : 0 < x
⊢ ∀ x ∈ {0, -1}ᶜ, x * (x + 1) ≠ 0