of_isBigO_im_re_rpow — Mathlib

English

For each k, the sequence n ↦ n.choose k is Θ(n^k).

Русский

Для каждого k последовательность n.choose k эквивалентна n^k по теории Θ.

LaTeX

$$$ \forall k, \ (\lambda n : \mathbb{N}) \mapsto (n.\mathrm{choose}\ k : \mathbb{R}) =_\Theta_{atTop} (\lambda n : \mathbb{N}) (n^k : \mathbb{R}) $$$

Lean4

theorem of_isBigO_im_re_rpow (hre : Tendsto re l atTop) (r : ℝ) (hr : im =O[l] fun z => z.re ^ r) : IsExpCmpFilter l :=
  ⟨hre, fun n =>
    IsLittleO.isBigO <|
      calc
        (fun z : ℂ => z.im ^ n) =O[l] fun z => (z.re ^ r) ^ n := hr.pow n
        _ =ᶠ[l] fun z => z.re ^ (r * n) :=
          ((hre.eventually_ge_atTop 0).mono fun z hz => by simp only [Real.rpow_mul hz r n, Real.rpow_natCast])
        _ =o[l] fun z => Real.exp z.re := (isLittleO_rpow_exp_atTop _).comp_tendsto hre⟩

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