openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /ImaginaryExponential.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SharedBinSupportCollision | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4FullFar | |
| open Set | |
| open scoped ContDiff | |
| noncomputable def imaginaryExponential (t : ℝ) : ℂ := Complex.exp (Complex.I*(t:ℂ)) | |
| theorem imaginaryExponential_contDiff : ContDiff ℝ ∞ imaginaryExponential := by | |
| unfold imaginaryExponential | |
| have hf : ContDiff ℝ ∞ (fun t : ℝ => (t:ℂ)) := Complex.ofRealCLM.contDiff | |
| fun_prop | |
| theorem imaginaryExponential_hasDerivAt (t : ℝ) : | |
| HasDerivAt imaginaryExponential (Complex.I*imaginaryExponential t) t := by | |
| unfold imaginaryExponential | |
| convert (((hasDerivAt_id t).ofReal_comp.const_mul Complex.I).cexp) using 1 <;> | |
| simp [mul_comm] | |
| theorem imaginaryExponential_iteratedDeriv (n : ℕ) : | |
| iteratedDeriv n imaginaryExponential = fun t => Complex.I^n * imaginaryExponential t := by | |
| induction n with | |
| | zero => simp | |
| | succ n ih => | |
| rw [iteratedDeriv_succ,ih] | |
| funext t | |
| rw [((imaginaryExponential_hasDerivAt t).const_mul (Complex.I^n)).deriv,pow_succ] | |
| ring | |
| theorem imaginaryExponential_derivative_norm (n : ℕ) (t : ℝ) : | |
| ‖iteratedDeriv n imaginaryExponential t‖ = 1 := by | |
| rw [imaginaryExponential_iteratedDeriv] | |
| simp [imaginaryExponential,Complex.norm_exp,Complex.mul_re] | |
| noncomputable def imaginaryTaylor (n : ℕ) (t : ℝ) : ℂ := | |
| ∑ k ∈ Finset.range (n+1), ((k.factorial:ℝ)⁻¹*t^k) • (Complex.I^k) | |
| theorem imaginaryExponential_taylor_bound (n : ℕ) {t : ℝ} (ht : 0 ≤ t) : | |
| ‖imaginaryExponential t - imaginaryTaylor n t‖ ≤ t^(n+1)/(n.factorial:ℝ) := by | |
| rcases eq_or_lt_of_le ht with rfl | ht | |
| · simp [imaginaryTaylor,imaginaryExponential,zero_pow_eq] | |
| have hd : UniqueDiffOn ℝ (Icc 0 t) := uniqueDiffOn_Icc ht | |
| have hs (k : ℕ) : ContDiff ℝ k imaginaryExponential := | |
| imaginaryExponential_contDiff.of_le (by exact_mod_cast (le_top : (k:ℕ∞) ≤ ⊤)) | |
| have hh := taylor_mean_remainder_bound ht.le (hs (n+1)).contDiffOn | |
| (show t ∈ Icc 0 t from ⟨ht.le,le_rfl⟩) (C := 1) (fun y hy => by | |
| rw [iteratedDerivWithin_eq_iteratedDeriv hd (hs (n+1)).contDiffAt hy, | |
| imaginaryExponential_derivative_norm]) | |
| have he : taylorWithinEval imaginaryExponential n (Icc 0 t) 0 t = imaginaryTaylor n t := by | |
| rw [taylor_within_apply] | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| rw [iteratedDerivWithin_eq_iteratedDeriv hd (hs k).contDiffAt (show (0:ℝ) ∈ Icc 0 t from ⟨le_rfl,ht.le⟩), | |
| imaginaryExponential_iteratedDeriv] | |
| simp [imaginaryExponential] | |
| simpa only [he,sub_zero,one_mul] using hh | |
| end Attempt4FullFar | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |