q : ℍ
hq : q.re = 0
⊢ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q
q : ℍ
hq : q.re = 0
⊢ exp ℝ q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q
rw [exp_eq_tsum]
q : ℍ
hq : q.re = 0
⊢ HasSum (fun n => (↑n !)⁻¹ • q ^ n) (↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q)
q : ℍ
hq : q.re = 0
⊢ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ • x ^ n) q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q
refine' HasSum.tsum_eq _
q : ℍ
hq : q.re = 0
⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => q) (↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q)
q : ℍ
hq : q.re = 0
⊢ HasSum (fun n => (↑n !)⁻¹ • q ^ n) (↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q)
simp_rw [← expSeries_apply_eq]
no goals
q : ℍ
hq : q.re = 0
⊢ HasSum (fun n => ↑(expSeries ℝ ℍ n) fun x => q) (↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q)
exact hasSum_expSeries_of_imaginary hq (Real.hasSum_cos _) (Real.hasSum_sin _)