exp_of_re_eq_zero — Mathlib

English

Analogous HasSum result for the imaginary case with another decomposition of the exponential series.

Русский

Аналогичный результат суммы expSeries для мнимого случая с другой декомпозицией экспонентной серии.

LaTeX

$$$\text{HasSum}\;\big(n \mapsto \expSeries_{\mathbb{R}}(\mathbb{H}, q)(n)\big) \; = \; \text{(real part)} + \text{(imaginary part)}.$$$

Lean4

/-- The closed form for the quaternion exponential on imaginary quaternions. -/
theorem exp_of_re_eq_zero (q : Quaternion ℝ) (hq : q.re = 0) : exp ℝ q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q :=
  by
  rw [exp_eq_tsum]
  refine HasSum.tsum_eq ?_
  simp_rw [← expSeries_apply_eq]
  exact hasSum_expSeries_of_imaginary hq (Real.hasSum_cos _) (Real.hasSum_sin _)

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