. norm_cast
  simp_rw [← Complex.real_smul]
  exact Function.hasTemperateGrowth_neg.cexp_real_smul_I
case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
⊢ Function.HasTemperateGrowth fun x => cexp (-↑x * I)

case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
⊢ Function.HasTemperateGrowth fun x => cexp (↑x * I)
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
⊢ Function.HasTemperateGrowth fun x => cexp (↑x * I)