| import OAI.NumberTheory.TwoPointCorrelations.HalaszPhaseCells | |
| namespace OAI | |
| /-! Exact primitive of the Archimedean phase, including the zero endpoint. -/ | |
| namespace TwoPointCorrelations | |
| open MeasureTheory | |
| lemma halasz_power_phase_eq_cpow (t x : ℝ) (hx : 0 < x) : | |
| halaszPowerPhase t x = (x : ℂ) ^ ((-t : ℂ) * Complex.I) := by | |
| rw [Complex.cpow_def_of_ne_zero (by exact_mod_cast hx.ne'), | |
| ← Complex.ofReal_log hx.le] | |
| unfold halaszPowerPhase | |
| congr 1 | |
| push_cast | |
| ring | |
| lemma halasz_power_phase_cpow_primitive (t x : ℝ) (hx : 0 ≤ x) : | |
| (x : ℂ) ^ ((-t : ℂ) * Complex.I + 1) = (x : ℂ) * halaszPowerPhase t x := by | |
| by_cases hx0 : x = 0 | |
| · subst x | |
| simp only [Complex.ofReal_zero, zero_mul] | |
| apply Complex.zero_cpow | |
| intro h | |
| have hr := congrArg Complex.re h | |
| norm_num [Complex.mul_re] at hr | |
| · have hxp : 0 < x := lt_of_le_of_ne hx (Ne.symm hx0) | |
| rw [Complex.cpow_add _ _ (by exact_mod_cast hx0), Complex.cpow_one, | |
| halasz_power_phase_eq_cpow t x hxp, mul_comm] | |
| lemma halasz_power_phase_integral (t a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) : | |
| (∫ x in a..b, halaszPowerPhase t x) = | |
| ((b : ℂ) * halaszPowerPhase t b - (a : ℂ) * halaszPowerPhase t a) / | |
| (1 + (-t : ℂ) * Complex.I) := by | |
| have he : (∫ x in a..b, halaszPowerPhase t x) = | |
| ∫ x : ℝ in a..b, (x : ℂ) ^ ((-t : ℂ) * Complex.I) := by | |
| apply intervalIntegral.integral_congr_uIoo | |
| intro x hx | |
| exact halasz_power_phase_eq_cpow t x | |
| (lt_of_le_of_lt (le_min ha hb) hx.1) | |
| rw [he, integral_cpow (Or.inl (by norm_num [Complex.mul_re])), | |
| halasz_power_phase_cpow_primitive t b hb, | |
| halasz_power_phase_cpow_primitive t a ha, add_comm ((-t : ℂ) * Complex.I) 1] | |
| lemma halasz_power_phase_intervalIntegrable (t a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) : | |
| IntervalIntegrable (halaszPowerPhase t) volume a b := by | |
| apply (intervalIntegral.intervalIntegrable_cpow' (r := (-t : ℂ) * Complex.I) | |
| (by norm_num [Complex.mul_re])).congr_uIoo | |
| intro x hx | |
| exact (halasz_power_phase_eq_cpow t x | |
| (lt_of_le_of_lt (le_min ha hb) hx.1)).symm | |
| lemma halasz_power_phase_integral_norm (t a b : ℝ) : | |
| ‖∫ x in a..b, halaszPowerPhase t x‖ ≤ |b - a| := by | |
| simpa using intervalIntegral.norm_integral_le_of_norm_le_const | |
| (a := a) (b := b) (C := 1) (f := halaszPowerPhase t) | |
| (fun x _ => (halasz_power_phase_norm t x).le) | |
| end TwoPointCorrelations | |
| end OAI | |