English
For every g ∈ GL(2, ℝ), the map τ ↦ num(g, τ) is MDifferentiable on the model with corners with Complex structure.
Русский
Для каждого g ∈ GL(2, ℝ) отображение τ ↦ num(g, τ) MD-дифференцируемо на модели с краями и комплексной структурой.
LaTeX
$$$MDifferentiable (modelWithCornersSelf \mathbb{C} \mathbb{C}) (modelWithCornersSelf \mathbb{C} \mathbb{C}) (\lambda τ, \; \mathrm{num}(g, τ))$$$
Lean4
theorem exp_half_dist (z w : ℍ) :
exp (dist z w / 2) = (dist (z : ℂ) w + dist (z : ℂ) (conj ↑w)) / (2 * √(z.im * w.im)) := by
rw [← sinh_add_cosh, sinh_half_dist, cosh_half_dist, add_div]