Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
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import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.complex.removable_singularity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Metric S... | Mathlib/Analysis/Complex/RemovableSingularity.lean | 46 | 57 | theorem differentiableOn_compl_singleton_and_continuousAt_iff {f : ℂ → E} {s : Set ℂ} {c : ℂ}
(hs : s ∈ 𝓝 c) :
DifferentiableOn ℂ f (s \ {c}) ∧ ContinuousAt f c ↔ DifferentiableOn ℂ f s := by |
refine ⟨?_, fun hd => ⟨hd.mono diff_subset, (hd.differentiableAt hs).continuousAt⟩⟩
rintro ⟨hd, hc⟩ x hx
rcases eq_or_ne x c with (rfl | hne)
· refine (analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt
?_ hc).differentiableAt.differentiableWithinAt
refine eventually_nhdsWithin_iff.2 ((ev... | 2,340 |
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.complex.removable_singularity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Metric S... | Mathlib/Analysis/Complex/RemovableSingularity.lean | 71 | 87 | theorem differentiableOn_update_limUnder_of_isLittleO {f : ℂ → E} {s : Set ℂ} {c : ℂ} (hc : s ∈ 𝓝 c)
(hd : DifferentiableOn ℂ f (s \ {c}))
(ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹) :
DifferentiableOn ℂ (update f c (limUnder (𝓝[≠] c) f)) s := by |
set F : ℂ → E := fun z => (z - c) • f z
suffices DifferentiableOn ℂ F (s \ {c}) ∧ ContinuousAt F c by
rw [differentiableOn_compl_singleton_and_continuousAt_iff hc, ← differentiableOn_dslope hc,
dslope_sub_smul] at this
have hc : Tendsto f (𝓝[≠] c) (𝓝 (deriv F c)) :=
continuousAt_update_same.m... | 2,340 |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 50 | 64 | theorem norm_cderiv_le (hr : 0 < r) (hf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M) :
‖cderiv r f z‖ ≤ M / r := by |
have hM : 0 ≤ M := by
obtain ⟨w, hw⟩ : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
exact (norm_nonneg _).trans (hf w hw)
have h1 : ∀ w ∈ sphere z r, ‖((w - z) ^ 2)⁻¹ • f w‖ ≤ M / r ^ 2 := by
intro w hw
simp only [mem_sphere_iff_norm, norm_eq_abs] at hw
simp only [norm_smul, i... | 2,341 |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 67 | 76 | theorem cderiv_sub (hr : 0 < r) (hf : ContinuousOn f (sphere z r))
(hg : ContinuousOn g (sphere z r)) : cderiv r (f - g) z = cderiv r f z - cderiv r g z := by |
have h1 : ContinuousOn (fun w : ℂ => ((w - z) ^ 2)⁻¹) (sphere z r) := by
refine ((continuous_id'.sub continuous_const).pow 2).continuousOn.inv₀ fun w hw h => hr.ne ?_
rwa [mem_sphere_iff_norm, sq_eq_zero_iff.mp h, norm_zero] at hw
simp_rw [cderiv, ← smul_sub]
congr 1
simpa only [Pi.sub_apply, smul_sub]... | 2,341 |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 79 | 86 | theorem norm_cderiv_lt (hr : 0 < r) (hfM : ∀ w ∈ sphere z r, ‖f w‖ < M)
(hf : ContinuousOn f (sphere z r)) : ‖cderiv r f z‖ < M / r := by |
obtain ⟨L, hL1, hL2⟩ : ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L := by
have e1 : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
have e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r) := continuous_norm.comp_continuousOn hf
obtain ⟨x, hx, hx'⟩ := (isCompact_sphere z r).exists_isMaxOn e1 e2
ex... | 2,341 |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 95 | 110 | theorem _root_.TendstoUniformlyOn.cderiv (hF : TendstoUniformlyOn F f φ (cthickening δ K))
(hδ : 0 < δ) (hFn : ∀ᶠ n in φ, ContinuousOn (F n) (cthickening δ K)) :
TendstoUniformlyOn (cderiv δ ∘ F) (cderiv δ f) φ K := by |
rcases φ.eq_or_neBot with rfl | hne
· simp only [TendstoUniformlyOn, eventually_bot, imp_true_iff]
have e1 : ContinuousOn f (cthickening δ K) := TendstoUniformlyOn.continuousOn hF hFn
rw [tendstoUniformlyOn_iff] at hF ⊢
rintro ε hε
filter_upwards [hF (ε * δ) (mul_pos hε hδ), hFn] with n h h' z hz
simp_rw... | 2,341 |
import Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence
import Mathlib.Analysis.Complex.UpperHalfPlane.Manifold
import Mathlib.Analysis.Complex.LocallyUniformLimit
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open ModularForm EisensteinSeries UpperHalfPlane Set Filter... | Mathlib/NumberTheory/ModularForms/EisensteinSeries/MDifferentiable.lean | 54 | 65 | theorem eisensteinSeries_SIF_MDifferentiable {k : ℤ} {N : ℕ} (hk : 3 ≤ k) (a : Fin 2 → ZMod N) :
MDifferentiable 𝓘(ℂ) 𝓘(ℂ) (eisensteinSeries_SIF a k) := by |
intro τ
suffices DifferentiableAt ℂ (↑ₕeisensteinSeries_SIF a k) τ.1 by
convert MDifferentiableAt.comp τ (DifferentiableAt.mdifferentiableAt this) τ.mdifferentiable_coe
exact funext fun z ↦ (comp_ofComplex (eisensteinSeries_SIF a k) z).symm
refine DifferentiableOn.differentiableAt ?_
((isOpen_lt cont... | 2,342 |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 65 | 88 | theorem schwarz_aux {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball c R₁))
(h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
‖dslope f c z‖ ≤ R₂ / R₁ := by |
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
suffices ∀ᶠ r in 𝓝[<] R₁, ‖dslope f c z‖ ≤ R₂ / r by
refine ge_of_tendsto ?_ this
exact (tendsto_const_nhds.div tendsto_id hR₁.ne').mono_left nhdsWithin_le_nhds
rw [mem_ball] at hz
filter_upwards [Ioo_mem_nhdsWithin_Iio ⟨hz, le_rfl⟩] with r hr
have hr₀ : ... | 2,343 |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 92 | 108 | theorem norm_dslope_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
(h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
‖dslope f c z‖ ≤ R₂ / R₁ := by |
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
have hR₂ : 0 < R₂ := nonempty_ball.1 ⟨f z, h_maps hz⟩
rcases eq_or_ne (dslope f c z) 0 with hc | hc
· rw [hc, norm_zero]; exact div_nonneg hR₂.le hR₁.le
rcases exists_dual_vector ℂ _ hc with ⟨g, hg, hgf⟩
have hg' : ‖g‖₊ = 1 := NNReal.eq hg
have hg₀ : ‖g‖₊ ≠ 0 ... | 2,343 |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 113 | 130 | theorem affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div [CompleteSpace E] [StrictConvexSpace ℝ E]
(hd : DifferentiableOn ℂ f (ball c R₁)) (h_maps : Set.MapsTo f (ball c R₁) (ball (f c) R₂))
(h_z₀ : z₀ ∈ ball c R₁) (h_eq : ‖dslope f c z₀‖ = R₂ / R₁) :
Set.EqOn f (fun z => f c + (z - c) • dslope f c z₀) (b... |
set g := dslope f c
rintro z hz
by_cases h : z = c; · simp [h]
have h_R₁ : 0 < R₁ := nonempty_ball.mp ⟨_, h_z₀⟩
have g_le_div : ∀ z ∈ ball c R₁, ‖g z‖ ≤ R₂ / R₁ := fun z hz =>
norm_dslope_le_div_of_mapsTo_ball hd h_maps hz
have g_max : IsMaxOn (norm ∘ g) (ball c R₁) z₀ :=
isMaxOn_iff.mpr fun z hz =... | 2,343 |
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
open Complex Set
open scoped Topology
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E]
variable {f g : E →... | Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean | 24 | 25 | theorem analyticOn_cexp : AnalyticOn ℂ exp univ := by |
rw [analyticOn_univ_iff_differentiable]; exact differentiable_exp
| 2,344 |
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
open Complex Set
open scoped Topology
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E]
variable {f g : E →... | Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean | 40 | 44 | theorem analyticAt_clog (m : z ∈ slitPlane) : AnalyticAt ℂ log z := by |
rw [analyticAt_iff_eventually_differentiableAt]
filter_upwards [isOpen_slitPlane.eventually_mem m]
intro z m
exact differentiableAt_id.clog m
| 2,344 |
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
open Complex Set
open scoped Topology
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E]
variable {f g : E →... | Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean | 57 | 64 | theorem AnalyticAt.cpow (fa : AnalyticAt ℂ f x) (ga : AnalyticAt ℂ g x)
(m : f x ∈ slitPlane) : AnalyticAt ℂ (fun z ↦ f z ^ g z) x := by |
have e : (fun z ↦ f z ^ g z) =ᶠ[𝓝 x] fun z ↦ exp (log (f z) * g z) := by
filter_upwards [(fa.continuousAt.eventually_ne (slitPlane_ne_zero m))]
intro z fz
simp only [fz, cpow_def, if_false]
rw [analyticAt_congr e]
exact ((fa.clog m).mul ga).cexp
| 2,344 |
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable section... | Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | 95 | 103 | theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
HasFDerivAt polarCoord.symm
(LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
!![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])) p := by |
rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]
convert HasFDerivAt.prod (𝕜 := ℝ)
(hasFDerivAt_fst.mul ((hasDerivAt_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))
(hasFDerivAt_fst.mul ((hasDerivAt_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using 2 <;>
simp [smul_smul, add_comm, neg_mul, smul_neg, n... | 2,345 |
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable section... | Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | 110 | 123 | theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by |
have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
intro x hx
simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
Classical.not_not] at hx
exact hx.2
have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
a... | 2,345 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.Alge... | Mathlib/Analysis/Fourier/FourierTransform.lean | 84 | 92 | theorem fourierIntegral_smul_const (e : AddChar 𝕜 𝕊) (μ : Measure V)
(L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (r : ℂ) :
fourierIntegral e μ L (r • f) = r • fourierIntegral e μ L f := by |
ext1 w
-- Porting note: was
-- simp only [Pi.smul_apply, fourierIntegral, smul_comm _ r, integral_smul]
simp only [Pi.smul_apply, fourierIntegral, ← integral_smul]
congr 1 with v
rw [smul_comm]
| 2,346 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.Alge... | Mathlib/Analysis/Fourier/FourierTransform.lean | 96 | 100 | theorem norm_fourierIntegral_le_integral_norm (e : AddChar 𝕜 𝕊) (μ : Measure V)
(L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (w : W) :
‖fourierIntegral e μ L f w‖ ≤ ∫ v : V, ‖f v‖ ∂μ := by |
refine (norm_integral_le_integral_norm _).trans (le_of_eq ?_)
simp_rw [norm_circle_smul]
| 2,346 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.Alge... | Mathlib/Analysis/Fourier/FourierTransform.lean | 104 | 114 | theorem fourierIntegral_comp_add_right [MeasurableAdd V] (e : AddChar 𝕜 𝕊) (μ : Measure V)
[μ.IsAddRightInvariant] (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (v₀ : V) :
fourierIntegral e μ L (f ∘ fun v ↦ v + v₀) =
fun w ↦ e (L v₀ w) • fourierIntegral e μ L f w := by |
ext1 w
dsimp only [fourierIntegral, Function.comp_apply, Submonoid.smul_def]
conv in L _ => rw [← add_sub_cancel_right v v₀]
rw [integral_add_right_eq_self fun v : V ↦ (e (-L (v - v₀) w) : ℂ) • f v, ← integral_smul]
congr 1 with v
rw [← smul_assoc, smul_eq_mul, ← Submonoid.coe_mul, ← e.map_add_eq_mul, ← Li... | 2,346 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.Alge... | Mathlib/Analysis/Fourier/FourierTransform.lean | 132 | 147 | theorem fourierIntegral_convergent_iff (he : Continuous e)
(hL : Continuous fun p : V × W ↦ L p.1 p.2) {f : V → E} (w : W) :
Integrable (fun v : V ↦ e (-L v w) • f v) μ ↔ Integrable f μ := by |
-- first prove one-way implication
have aux {g : V → E} (hg : Integrable g μ) (x : W) :
Integrable (fun v : V ↦ e (-L v x) • g v) μ := by
have c : Continuous fun v ↦ e (-L v x) :=
he.comp (hL.comp (continuous_prod_mk.mpr ⟨continuous_id, continuous_const⟩)).neg
simp_rw [← integrable_norm_iff (c.... | 2,346 |
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 56 | 103 | theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
(hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖)
(m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m := by |
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩
have neK : ∀ (K : Comp... | 2,347 |
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 107 | 121 | theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
(h_norm :
∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖)
(h_sum : Summable fun n : ℤ => 𝓕 f n) (x : ℝ) :
∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 f n * fourier n (x : UnitAddCircle) := by |
let F : C(UnitAddCircle, ℂ) :=
⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this x).tsum_eq.symm usin... | 2,347 |
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 131 | 157 | theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : f =O[atTop] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
(fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atTop] fun x : ℝ => |x| ^ (-b) := by |
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved in-line rather than being formulated as a separate lemma.
have claim : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y →
... | 2,347 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 64 | 83 | theorem integral_comp_smul (f : E → F) (R : ℝ) :
∫ x, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x, f x ∂μ := by |
by_cases hF : CompleteSpace F; swap
· simp [integral, hF]
rcases eq_or_ne R 0 with (rfl | hR)
· simp only [zero_smul, integral_const]
rcases Nat.eq_zero_or_pos (finrank ℝ E) with (hE | hE)
· have : Subsingleton E := finrank_zero_iff.1 hE
have : f = fun _ => f 0 := by ext x; rw [Subsingleton.elim ... | 2,348 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 89 | 91 | theorem integral_comp_smul_of_nonneg (f : E → F) (R : ℝ) {hR : 0 ≤ R} :
∫ x, f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ x, f x ∂μ := by |
rw [integral_comp_smul μ f R, abs_of_nonneg (inv_nonneg.2 (pow_nonneg hR _))]
| 2,348 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 97 | 99 | theorem integral_comp_inv_smul (f : E → F) (R : ℝ) :
∫ x, f (R⁻¹ • x) ∂μ = |R ^ finrank ℝ E| • ∫ x, f x ∂μ := by |
rw [integral_comp_smul μ f R⁻¹, inv_pow, inv_inv]
| 2,348 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 105 | 107 | theorem integral_comp_inv_smul_of_nonneg (f : E → F) {R : ℝ} (hR : 0 ≤ R) :
∫ x, f (R⁻¹ • x) ∂μ = R ^ finrank ℝ E • ∫ x, f x ∂μ := by |
rw [integral_comp_inv_smul μ f R, abs_of_nonneg (pow_nonneg hR _)]
| 2,348 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 110 | 123 | theorem setIntegral_comp_smul (f : E → F) {R : ℝ} (s : Set E) (hR : R ≠ 0) :
∫ x in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ := by |
let e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
calc
∫ x in s, f (R • x) ∂μ
= ∫ x in e ⁻¹' (e.symm ⁻¹' s), f (e x) ∂μ := by simp [← preimage_comp]; rfl
_ = ∫ y in e.symm ⁻¹' s, f y ∂map (fun x ↦ R • x) μ := (setIntegral_map_equiv _ _ _).symm
_ = |(R ^ finrank ℝ E)⁻¹| • ∫ y in e.sym... | 2,348 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 128 | 130 | theorem setIntegral_comp_smul_of_pos (f : E → F) {R : ℝ} (s : Set E) (hR : 0 < R) :
∫ x in s, f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ x in R • s, f x ∂μ := by |
rw [setIntegral_comp_smul μ f s hR.ne', abs_of_nonneg (inv_nonneg.2 (pow_nonneg hR.le _))]
| 2,348 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 135 | 137 | theorem integral_comp_mul_left (g : ℝ → F) (a : ℝ) :
(∫ x : ℝ, g (a * x)) = |a⁻¹| • ∫ y : ℝ, g y := by |
simp_rw [← smul_eq_mul, Measure.integral_comp_smul, FiniteDimensional.finrank_self, pow_one]
| 2,348 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 163 | 177 | theorem integrable_comp_smul_iff {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ]
(f : E → F) {R : ℝ} (hR : R ≠ 0) : Integrable (fun x => f (R • x)) μ ↔ Integrable f μ := by |
-- reduce to one-way implication
suffices
∀ {g : E → F} (_ : Integrable g μ) {S : ℝ} (_ : S ≠ 0), Integrable (fun x => g (S • x)) μ by
refine ⟨fun hf => ?_, fun hf => this hf hR⟩
convert this hf (inv_ne_zero hR)
rw [← mul_smul, mul_inv_cancel hR, one_smul]
-- now prove
intro g hg S hS
let t :... | 2,348 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 47 | 50 | theorem MellinConvergent.const_smul {f : ℝ → E} {s : ℂ} (hf : MellinConvergent f s) {𝕜 : Type*}
[NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] (c : 𝕜) :
MellinConvergent (fun t => c • f t) s := by |
simpa only [MellinConvergent, smul_comm] using hf.smul c
| 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 53 | 56 | theorem MellinConvergent.cpow_smul {f : ℝ → E} {s a : ℂ} :
MellinConvergent (fun t => (t : ℂ) ^ a • f t) s ↔ MellinConvergent f (s + a) := by |
refine integrableOn_congr_fun (fun t ht => ?_) measurableSet_Ioi
simp_rw [← sub_add_eq_add_sub, cpow_add _ _ (ofReal_ne_zero.2 <| ne_of_gt ht), mul_smul]
| 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 64 | 75 | theorem MellinConvergent.comp_mul_left {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : 0 < a) :
MellinConvergent (fun t => f (a * t)) s ↔ MellinConvergent f s := by |
have := integrableOn_Ioi_comp_mul_left_iff (fun t : ℝ => (t : ℂ) ^ (s - 1) • f t) 0 ha
rw [mul_zero] at this
have h1 : EqOn (fun t : ℝ => (↑(a * t) : ℂ) ^ (s - 1) • f (a * t))
((a : ℂ) ^ (s - 1) • fun t : ℝ => (t : ℂ) ^ (s - 1) • f (a * t)) (Ioi 0) := fun t ht ↦ by
simp only [ofReal_mul, mul_cpow_ofRea... | 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 78 | 87 | theorem MellinConvergent.comp_rpow {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : a ≠ 0) :
MellinConvergent (fun t => f (t ^ a)) s ↔ MellinConvergent f (s / a) := by |
refine Iff.trans ?_ (integrableOn_Ioi_comp_rpow_iff' _ ha)
rw [MellinConvergent]
refine integrableOn_congr_fun (fun t ht => ?_) measurableSet_Ioi
dsimp only [Pi.smul_apply]
rw [← Complex.coe_smul (t ^ (a - 1)), ← mul_smul, ← cpow_mul_ofReal_nonneg (le_of_lt ht),
ofReal_cpow (le_of_lt ht), ← cpow_add _ _ ... | 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 106 | 109 | theorem mellin_cpow_smul (f : ℝ → E) (s a : ℂ) :
mellin (fun t => (t : ℂ) ^ a • f t) s = mellin f (s + a) := by |
refine setIntegral_congr measurableSet_Ioi fun t ht => ?_
simp_rw [← sub_add_eq_add_sub, cpow_add _ _ (ofReal_ne_zero.2 <| ne_of_gt ht), mul_smul]
| 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 112 | 114 | theorem mellin_const_smul (f : ℝ → E) (s : ℂ) {𝕜 : Type*} [NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] (c : 𝕜) :
mellin (fun t => c • f t) s = c • mellin f s := by | simp only [mellin, smul_comm, integral_smul]
| 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 117 | 118 | theorem mellin_div_const (f : ℝ → ℂ) (s a : ℂ) : mellin (fun t => f t / a) s = mellin f s / a := by |
simp_rw [mellin, smul_eq_mul, ← mul_div_assoc, integral_div]
| 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 196 | 205 | theorem mellin_convergent_iff_norm [NormedSpace ℂ E] {f : ℝ → E} {T : Set ℝ} (hT : T ⊆ Ioi 0)
(hT' : MeasurableSet T) (hfc : AEStronglyMeasurable f <| volume.restrict <| Ioi 0) {s : ℂ} :
IntegrableOn (fun t : ℝ => (t : ℂ) ^ (s - 1) • f t) T ↔
IntegrableOn (fun t : ℝ => t ^ (s.re - 1) * ‖f t‖) T := by |
have : AEStronglyMeasurable (fun t : ℝ => (t : ℂ) ^ (s - 1) • f t) (volume.restrict T) := by
refine ((ContinuousAt.continuousOn ?_).aestronglyMeasurable hT').smul (hfc.mono_set hT)
exact fun t ht => continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_gt (hT ht))
rw [IntegrableOn, ← integrable_norm_iff this... | 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 210 | 231 | theorem mellin_convergent_top_of_isBigO {f : ℝ → ℝ}
(hfc : AEStronglyMeasurable f <| volume.restrict (Ioi 0)) {a s : ℝ}
(hf : f =O[atTop] (· ^ (-a))) (hs : s < a) :
∃ c : ℝ, 0 < c ∧ IntegrableOn (fun t : ℝ => t ^ (s - 1) * f t) (Ioi c) := by |
obtain ⟨d, hd'⟩ := hf.isBigOWith
simp_rw [IsBigOWith, eventually_atTop] at hd'
obtain ⟨e, he⟩ := hd'
have he' : 0 < max e 1 := zero_lt_one.trans_le (le_max_right _ _)
refine ⟨max e 1, he', ?_, ?_⟩
· refine AEStronglyMeasurable.mul ?_ (hfc.mono_set (Ioi_subset_Ioi he'.le))
refine (ContinuousAt.continuou... | 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 237 | 264 | theorem mellin_convergent_zero_of_isBigO {b : ℝ} {f : ℝ → ℝ}
(hfc : AEStronglyMeasurable f <| volume.restrict (Ioi 0))
(hf : f =O[𝓝[>] 0] (· ^ (-b))) {s : ℝ} (hs : b < s) :
∃ c : ℝ, 0 < c ∧ IntegrableOn (fun t : ℝ => t ^ (s - 1) * f t) (Ioc 0 c) := by |
obtain ⟨d, _, hd'⟩ := hf.exists_pos
simp_rw [IsBigOWith, eventually_nhdsWithin_iff, Metric.eventually_nhds_iff, gt_iff_lt] at hd'
obtain ⟨ε, hε, hε'⟩ := hd'
refine ⟨ε, hε, integrableOn_Ioc_iff_integrableOn_Ioo.mpr ⟨?_, ?_⟩⟩
· refine AEStronglyMeasurable.mul ?_ (hfc.mono_set Ioo_subset_Ioi_self)
refine (C... | 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 304 | 312 | theorem isBigO_rpow_top_log_smul [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : b < a)
(hf : f =O[atTop] (· ^ (-a))) :
(fun t : ℝ => log t • f t) =O[atTop] (· ^ (-b)) := by |
refine
((isLittleO_log_rpow_atTop (sub_pos.mpr hab)).isBigO.smul hf).congr'
(eventually_of_forall fun t => by rfl)
((eventually_gt_atTop 0).mp (eventually_of_forall fun t ht => ?_))
simp only
rw [smul_eq_mul, ← rpow_add ht, ← sub_eq_add_neg, sub_eq_add_neg a, add_sub_cancel_left]
| 2,349 |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 317 | 332 | theorem isBigO_rpow_zero_log_smul [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : a < b)
(hf : f =O[𝓝[>] 0] (· ^ (-a))) :
(fun t : ℝ => log t • f t) =O[𝓝[>] 0] (· ^ (-b)) := by |
have : log =o[𝓝[>] 0] fun t : ℝ => t ^ (a - b) := by
refine ((isLittleO_log_rpow_atTop (sub_pos.mpr hab)).neg_left.comp_tendsto
tendsto_inv_zero_atTop).congr'
(eventually_nhdsWithin_iff.mpr <| eventually_of_forall fun t ht => ?_)
(eventually_nhdsWithin_iff.mpr <| eventually_of_forall fun t... | 2,349 |
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-community/mathlib"@"cca40788df1b8755d5baf17ab2f27dacc2e17acb"
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 56 | 67 | theorem Gamma_integrand_isLittleO (s : ℝ) :
(fun x : ℝ => exp (-x) * x ^ s) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) := by |
refine isLittleO_of_tendsto (fun x hx => ?_) ?_
· exfalso; exact (exp_pos (-(1 / 2) * x)).ne' hx
have : (fun x : ℝ => exp (-x) * x ^ s / exp (-(1 / 2) * x)) =
(fun x : ℝ => exp (1 / 2 * x) / x ^ s)⁻¹ := by
ext1 x
field_simp [exp_ne_zero, exp_neg, ← Real.exp_add]
left
ring
rw [this]
exac... | 2,350 |
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-community/mathlib"@"cca40788df1b8755d5baf17ab2f27dacc2e17acb"
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 71 | 82 | theorem GammaIntegral_convergent {s : ℝ} (h : 0 < s) :
IntegrableOn (fun x : ℝ => exp (-x) * x ^ (s - 1)) (Ioi 0) := by |
rw [← Ioc_union_Ioi_eq_Ioi (@zero_le_one ℝ _ _ _ _), integrableOn_union]
constructor
· rw [← integrableOn_Icc_iff_integrableOn_Ioc]
refine IntegrableOn.continuousOn_mul continuousOn_id.neg.rexp ?_ isCompact_Icc
refine (intervalIntegrable_iff_integrableOn_Icc_of_le zero_le_one).mp ?_
exact intervalInt... | 2,350 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 31 | 43 | theorem exp_neg_mul_rpow_isLittleO_exp_neg {p b : ℝ} (hb : 0 < b) (hp : 1 < p) :
(fun x : ℝ => exp (- b * x ^ p)) =o[atTop] fun x : ℝ => exp (-x) := by |
rw [isLittleO_exp_comp_exp_comp]
suffices Tendsto (fun x => x * (b * x ^ (p - 1) + -1)) atTop atTop by
refine Tendsto.congr' ?_ this
refine eventuallyEq_of_mem (Ioi_mem_atTop (0 : ℝ)) (fun x hx => ?_)
rw [mem_Ioi] at hx
rw [rpow_sub_one hx.ne']
field_simp [hx.ne']
ring
apply Tendsto.atTop... | 2,351 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 45 | 48 | theorem exp_neg_mul_sq_isLittleO_exp_neg {b : ℝ} (hb : 0 < b) :
(fun x : ℝ => exp (-b * x ^ 2)) =o[atTop] fun x : ℝ => exp (-x) := by |
simp_rw [← rpow_two]
exact exp_neg_mul_rpow_isLittleO_exp_neg hb one_lt_two
| 2,351 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 51 | 55 | theorem rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg (s : ℝ) {b p : ℝ} (hp : 1 < p) (hb : 0 < b) :
(fun x : ℝ => x ^ s * exp (- b * x ^ p)) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) := by |
apply ((isBigO_refl (fun x : ℝ => x ^ s) atTop).mul_isLittleO
(exp_neg_mul_rpow_isLittleO_exp_neg hb hp)).trans
simpa only [mul_comm] using Real.Gamma_integrand_isLittleO s
| 2,351 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 57 | 60 | theorem rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg {b : ℝ} (hb : 0 < b) (s : ℝ) :
(fun x : ℝ => x ^ s * exp (-b * x ^ 2)) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) := by |
simp_rw [← rpow_two]
exact rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg s one_lt_two hb
| 2,351 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 63 | 89 | theorem integrableOn_rpow_mul_exp_neg_rpow {p s : ℝ} (hs : -1 < s) (hp : 1 ≤ p) :
IntegrableOn (fun x : ℝ => x ^ s * exp (- x ^ p)) (Ioi 0) := by |
obtain hp | hp := le_iff_lt_or_eq.mp hp
· have h_exp : ∀ x, ContinuousAt (fun x => exp (- x)) x := fun x => continuousAt_neg.rexp
rw [← Ioc_union_Ioi_eq_Ioi zero_le_one, integrableOn_union]
constructor
· rw [← integrableOn_Icc_iff_integrableOn_Ioc]
refine IntegrableOn.mul_continuousOn ?_ ?_ isCom... | 2,351 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 91 | 102 | theorem integrableOn_rpow_mul_exp_neg_mul_rpow {p s b : ℝ} (hs : -1 < s) (hp : 1 ≤ p) (hb : 0 < b) :
IntegrableOn (fun x : ℝ => x ^ s * exp (- b * x ^ p)) (Ioi 0) := by |
have hib : 0 < b ^ (-p⁻¹) := rpow_pos_of_pos hb _
suffices IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * exp (-x ^ p))) (Ioi 0) by
rw [show 0 = b ^ (-p⁻¹) * 0 by rw [mul_zero], ← integrableOn_Ioi_comp_mul_left_iff _ _ hib]
refine this.congr_fun (fun _ hx => ?_) measurableSet_Ioi
rw [← mul_assoc, mu... | 2,351 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
set_option linter.uppercaseLean3 false
noncomputable section
open Filter Set MeasureTheory
open scoped Na... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 106 | 161 | theorem Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma {s t a b : ℝ} (hs : 0 < s) (ht : 0 < t)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :
Gamma (a * s + b * t) ≤ Gamma s ^ a * Gamma t ^ b := by |
-- We will apply Hölder's inequality, for the conjugate exponents `p = 1 / a`
-- and `q = 1 / b`, to the functions `f a s` and `f b t`, where `f` is as follows:
let f : ℝ → ℝ → ℝ → ℝ := fun c u x => exp (-c * x) * x ^ (c * (u - 1))
have e : IsConjExponent (1 / a) (1 / b) := Real.isConjExponent_one_div ha hb ha... | 2,352 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
set_option linter.uppercaseLean3 false
noncomputable section
open Filter Set MeasureTheory
open scoped Na... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 164 | 173 | theorem convexOn_log_Gamma : ConvexOn ℝ (Ioi 0) (log ∘ Gamma) := by |
refine convexOn_iff_forall_pos.mpr ⟨convex_Ioi _, fun x hx y hy a b ha hb hab => ?_⟩
have : b = 1 - a := by linarith
subst this
simp_rw [Function.comp_apply, smul_eq_mul]
simp only [mem_Ioi] at hx hy
rw [← log_rpow, ← log_rpow, ← log_mul]
· gcongr
exact Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma ... | 2,352 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 63 | 76 | theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by |
apply IntervalIntegrable.mul_continuousOn
· refine intervalIntegral.intervalIntegrable_cpow' ?_
rwa [sub_re, one_re, ← zero_sub, sub_lt_sub_iff_right]
· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : ℝ) ≤ 1 / 2)] at hx
apply ContinuousAt.cpow
· exact (continuo... | 2,353 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 80 | 90 | theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by |
refine (betaIntegral_convergent_left hu v).trans ?_
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> · push_cast; ring
· norm_num
· norm_num
| 2,353 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 93 | 102 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by |
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul,... | 2,353 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 105 | 111 | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by |
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
· rwa [sub_re, one_re, ← sub_pos, sub_neg_eq_add, sub_add_cancel]
| 2,353 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 114 | 132 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by |
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, ← intervalIntegral.int... | 2,353 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 136 | 151 | theorem Gamma_mul_Gamma_eq_betaIntegral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t := by |
-- Note that we haven't proved (yet) that the Gamma function has no zeroes, so we can't formulate
-- this as a formula for the Beta function.
have conv_int := integral_posConvolution
(GammaIntegral_convergent hs) (GammaIntegral_convergent ht) (ContinuousLinearMap.mul ℝ ℂ)
simp_rw [ContinuousLinearMap.mul_a... | 2,353 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 530 | 534 | theorem one_div_Gamma_eq_self_mul_one_div_Gamma_add_one (s : ℂ) :
(Gamma s)⁻¹ = s * (Gamma (s + 1))⁻¹ := by |
rcases ne_or_eq s 0 with (h | rfl)
· rw [Gamma_add_one s h, mul_inv, mul_inv_cancel_left₀ h]
· rw [zero_add, Gamma_zero, inv_zero, zero_mul]
| 2,353 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.PSeriesComplex
#align_import number_theory.zeta_function from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
o... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 103 | 105 | theorem completedRiemannZeta₀_one_sub (s : ℂ) :
completedRiemannZeta₀ (1 - s) = completedRiemannZeta₀ s := by |
rw [← completedHurwitzZetaEven₀_zero, ← completedCosZeta₀_zero, completedHurwitzZetaEven₀_one_sub]
| 2,354 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.PSeriesComplex
#align_import number_theory.zeta_function from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
o... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 110 | 112 | theorem completedRiemannZeta_one_sub (s : ℂ) :
completedRiemannZeta (1 - s) = completedRiemannZeta s := by |
rw [← completedHurwitzZetaEven_zero, ← completedCosZeta_zero, completedHurwitzZetaEven_one_sub]
| 2,354 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.PSeriesComplex
#align_import number_theory.zeta_function from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
o... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 149 | 150 | theorem riemannZeta_zero : riemannZeta 0 = -1 / 2 := by |
simp_rw [riemannZeta, hurwitzZetaEven, Function.update_same, if_true]
| 2,354 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.PSeriesComplex
#align_import number_theory.zeta_function from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
o... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 164 | 166 | theorem riemannZeta_one_sub {s : ℂ} (hs : ∀ n : ℕ, s ≠ -n) (hs' : s ≠ 1) :
riemannZeta (1 - s) = 2 * (2 * π) ^ (-s) * Gamma s * cos (π * s / 2) * riemannZeta s := by |
rw [riemannZeta, hurwitzZetaEven_one_sub 0 hs (Or.inr hs'), cosZeta_zero, hurwitzZetaEven_zero]
| 2,354 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.PSeriesComplex
#align_import number_theory.zeta_function from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
o... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 179 | 189 | theorem completedZeta_eq_tsum_of_one_lt_re {s : ℂ} (hs : 1 < re s) :
completedRiemannZeta s =
(π : ℂ) ^ (-s / 2) * Gamma (s / 2) * ∑' n : ℕ, 1 / (n : ℂ) ^ s := by |
have := (hasSum_nat_completedCosZeta 0 hs).tsum_eq.symm
simp only [QuotientAddGroup.mk_zero, completedCosZeta_zero] at this
simp only [this, Gammaℝ_def, mul_zero, zero_mul, Real.cos_zero, ofReal_one, mul_one, mul_one_div,
← tsum_mul_left]
congr 1 with n
split_ifs with h
· simp only [h, Nat.cast_zero, z... | 2,354 |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 91 | 94 | theorem riemannZeta_eulerProduct_hasProd (hs : 1 < s.re) :
HasProd (fun p : Primes ↦ (1 - (p : ℂ) ^ (-s))⁻¹) (riemannZeta s) := by |
rw [← tsum_riemannZetaSummand hs]
apply eulerProduct_completely_multiplicative_hasProd <| summable_riemannZetaSummand hs
| 2,355 |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 104 | 108 | theorem riemannZeta_eulerProduct (hs : 1 < s.re) :
Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - (p : ℂ) ^ (-s))⁻¹) atTop
(𝓝 (riemannZeta s)) := by |
rw [← tsum_riemannZetaSummand hs]
apply eulerProduct_completely_multiplicative <| summable_riemannZetaSummand hs
| 2,355 |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 114 | 118 | theorem dirichletLSeries_eulerProduct_hasProd {N : ℕ} (χ : DirichletCharacter ℂ N)
(hs : 1 < s.re) :
HasProd (fun p : Primes ↦ (1 - χ p * (p : ℂ) ^ (-s))⁻¹) (L ↗χ s) := by |
rw [← tsum_dirichletSummand χ hs]
convert eulerProduct_completely_multiplicative_hasProd <| summable_dirichletSummand χ hs
| 2,355 |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 129 | 133 | theorem dirichletLSeries_eulerProduct {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :
Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - χ p * (p : ℂ) ^ (-s))⁻¹) atTop
(𝓝 (L ↗χ s)) := by |
rw [← tsum_dirichletSummand χ hs]
apply eulerProduct_completely_multiplicative <| summable_dirichletSummand χ hs
| 2,355 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
| Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 49 | 67 | theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [← (hasSum_nat_cosZeta x (?_ : 1 < re (2 * k))).tsum_eq]
refine Eq.trans ?_ <| (congr_arg ofReal' (hasSum_one_div_nat_pow_mul_cos hk hx).tsum_eq).trans ?_
· rw [ofReal_tsum]
refine tsum_congr fun n ↦ ?_
rw [mul_comm (1 / _), mul_one_div, ofReal_div, mul_assoc (2 * π), mul_comm x n, ← mul_assoc,
... | 2,356 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 76 | 97 | theorem sinZeta_two_mul_nat_add_one (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [← (hasSum_nat_sinZeta x (?_ : 1 < re (2 * k + 1))).tsum_eq]
refine Eq.trans ?_ <| (congr_arg ofReal' (hasSum_one_div_nat_pow_mul_sin hk hx).tsum_eq).trans ?_
· rw [ofReal_tsum]
refine tsum_congr fun n ↦ ?_
rw [mul_comm (1 / _), mul_one_div, ofReal_div, mul_assoc (2 * π), mul_comm x n, ← mul_assoc]
... | 2,356 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 100 | 110 | theorem cosZeta_two_mul_nat' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) / (2 * k) / Gammaℂ (2 * k) *
((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [cosZeta_two_mul_nat hk hx]
congr 1
have : (2 * k)! = (2 * k) * Complex.Gamma (2 * k) := by
rw [(by { norm_cast; omega } : 2 * (k : ℂ) = ↑(2 * k - 1) + 1), Complex.Gamma_nat_eq_factorial,
← Nat.cast_add_one, ← Nat.cast_mul, ← Nat.factorial_succ, Nat.sub_add_cancel (by omega)]
simp_rw [this, Gammaℂ... | 2,356 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 113 | 124 | theorem sinZeta_two_mul_nat_add_one' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) / (2 * k + 1) / Gammaℂ (2 * k + 1) *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [sinZeta_two_mul_nat_add_one hk hx]
congr 1
have : (2 * k + 1)! = (2 * k + 1) * Complex.Gamma (2 * k + 1) := by
rw [(by simp : Complex.Gamma (2 * k + 1) = Complex.Gamma (↑(2 * k) + 1)),
Complex.Gamma_nat_eq_factorial, ← Nat.cast_ofNat (R := ℂ), ← Nat.cast_mul,
← Nat.cast_add_one, ← Nat.cast_m... | 2,356 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 126 | 146 | theorem hurwitzZetaEven_one_sub_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
hurwitzZetaEven x (1 - 2 * k) =
-1 / (2 * k) * ((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
have h1 (n : ℕ) : (2 * k : ℂ) ≠ -n := by
rw [← Int.cast_ofNat, ← Int.cast_natCast, ← Int.cast_mul, ← Int.cast_natCast n, ← Int.cast_neg,
Ne, Int.cast_inj, ← Ne]
refine ne_of_gt ((neg_nonpos_of_nonneg n.cast_nonneg).trans_lt (mul_pos two_pos ?_))
exact Nat.cast_pos.mpr (Nat.pos_of_ne_zero hk)
have... | 2,356 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
| Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 211 | 217 | theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (2 * k)! := by |
convert congr_arg ((↑) : ℝ → ℂ) (hasSum_zeta_nat hk).tsum_eq
· rw [← Nat.cast_two, ← Nat.cast_mul, zeta_nat_eq_tsum_of_gt_one (by omega)]
simp only [push_cast]
· norm_cast
| 2,356 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 220 | 224 | theorem riemannZeta_two : riemannZeta 2 = (π : ℂ) ^ 2 / 6 := by |
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_two.tsum_eq
· rw [← Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two]
simp only [push_cast]
· norm_cast
| 2,356 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 227 | 232 | theorem riemannZeta_four : riemannZeta 4 = π ^ 4 / 90 := by |
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_four.tsum_eq
· rw [← Nat.cast_one, show (4 : ℂ) = (4 : ℕ) by norm_num,
zeta_nat_eq_tsum_of_gt_one (by norm_num : 1 < 4)]
simp only [push_cast]
· norm_cast
| 2,356 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 51 | 55 | theorem norm_cexp_neg_mul_sq_add_mul_I (b : ℂ) (c T : ℝ) :
‖cexp (-b * (T + c * I) ^ 2)‖ = exp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2)) := by |
rw [Complex.norm_eq_abs, Complex.abs_exp, neg_mul, neg_re, ← re_add_im b]
simp only [sq, re_add_im, mul_re, mul_im, add_re, add_im, ofReal_re, ofReal_im, I_re, I_im]
ring_nf
| 2,357 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 59 | 66 | theorem norm_cexp_neg_mul_sq_add_mul_I' (hb : b.re ≠ 0) (c T : ℝ) :
‖cexp (-b * (T + c * I) ^ 2)‖ =
exp (-(b.re * (T - b.im * c / b.re) ^ 2 - c ^ 2 * (b.im ^ 2 / b.re + b.re))) := by |
have :
b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2 =
b.re * (T - b.im * c / b.re) ^ 2 - c ^ 2 * (b.im ^ 2 / b.re + b.re) := by
field_simp; ring
rw [norm_cexp_neg_mul_sq_add_mul_I, this]
| 2,357 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 70 | 112 | theorem verticalIntegral_norm_le (hb : 0 < b.re) (c : ℝ) {T : ℝ} (hT : 0 ≤ T) :
‖verticalIntegral b c T‖ ≤
(2 : ℝ) * |c| * exp (-(b.re * T ^ 2 - (2 : ℝ) * |b.im| * |c| * T - b.re * c ^ 2)) := by |
-- first get uniform bound for integrand
have vert_norm_bound :
∀ {T : ℝ},
0 ≤ T →
∀ {c y : ℝ},
|y| ≤ |c| →
‖cexp (-b * (T + y * I) ^ 2)‖ ≤
exp (-(b.re * T ^ 2 - (2 : ℝ) * |b.im| * |c| * T - b.re * c ^ 2)) := by
intro T hT c y hy
rw [norm_cexp_neg_mul_s... | 2,357 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 115 | 129 | theorem tendsto_verticalIntegral (hb : 0 < b.re) (c : ℝ) :
Tendsto (verticalIntegral b c) atTop (𝓝 0) := by |
-- complete proof using squeeze theorem:
rw [tendsto_zero_iff_norm_tendsto_zero]
refine
tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds ?_
(eventually_of_forall fun _ => norm_nonneg _)
((eventually_ge_atTop (0 : ℝ)).mp
(eventually_of_forall fun T hT => verticalIntegral_norm_... | 2,357 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 132 | 145 | theorem integrable_cexp_neg_mul_sq_add_real_mul_I (hb : 0 < b.re) (c : ℝ) :
Integrable fun x : ℝ => cexp (-b * (x + c * I) ^ 2) := by |
refine
⟨(Complex.continuous_exp.comp
(continuous_const.mul
((continuous_ofReal.add continuous_const).pow 2))).aestronglyMeasurable,
?_⟩
rw [← hasFiniteIntegral_norm_iff]
simp_rw [norm_cexp_neg_mul_sq_add_mul_I' hb.ne', neg_sub _ (c ^ 2 * _),
sub_eq_add_neg _ (b.re * _), Real.e... | 2,357 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Analysis.Fourier.PoissonSummation
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform RealInnerProductSpace
open Complex hiding exp continuous_exp abs_of_nonneg sq_abs
noncomp... | Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean | 68 | 76 | theorem tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact {a : ℝ} (ha : 0 < a) (s : ℝ) :
Tendsto (fun x : ℝ => |x| ^ s * rexp (-a * x ^ 2)) (cocompact ℝ) (𝓝 0) := by |
conv in rexp _ => rw [← sq_abs]
erw [cocompact_eq_atBot_atTop, ← comap_abs_atTop,
@tendsto_comap'_iff _ _ _ (fun y => y ^ s * rexp (-a * y ^ 2)) _ _ _
(mem_atTop_sets.mpr ⟨0, fun b hb => ⟨b, abs_of_nonneg hb⟩⟩)]
exact
(rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg ha s).tendsto_zero_of_tendsto
(t... | 2,358 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Analysis.Fourier.PoissonSummation
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform RealInnerProductSpace
open Complex hiding exp continuous_exp abs_of_nonneg sq_abs
noncomp... | Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean | 79 | 83 | theorem isLittleO_exp_neg_mul_sq_cocompact {a : ℂ} (ha : 0 < a.re) (s : ℝ) :
(fun x : ℝ => Complex.exp (-a * x ^ 2)) =o[cocompact ℝ] fun x : ℝ => |x| ^ s := by |
convert cexp_neg_quadratic_isLittleO_abs_rpow_cocompact (?_ : (-a).re < 0) 0 s using 1
· simp_rw [zero_mul, add_zero]
· rwa [neg_re, neg_lt_zero]
| 2,358 |
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Analysis.Fourier.PoissonSummation
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform RealInnerProductSpace
open Complex hiding exp continuous_exp abs_of_nonneg sq_abs
noncomp... | Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean | 88 | 122 | theorem Complex.tsum_exp_neg_quadratic {a : ℂ} (ha : 0 < a.re) (b : ℂ) :
(∑' n : ℤ, cexp (-π * a * n ^ 2 + 2 * π * b * n)) =
1 / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n + I * b) ^ 2) := by |
let f : ℝ → ℂ := fun x ↦ cexp (-π * a * x ^ 2 + 2 * π * b * x)
have hCf : Continuous f := by
refine Complex.continuous_exp.comp (Continuous.add ?_ ?_)
· exact continuous_const.mul (Complex.continuous_ofReal.pow 2)
· exact continuous_const.mul Complex.continuous_ofReal
have hFf : 𝓕 f = fun x : ℝ ↦ 1 ... | 2,358 |
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
open Real Set MeasureTheory MeasureTheory.Measure
section real
| Mathlib/MeasureTheory/Integral/Gamma.lean | 21 | 37 | theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q + 1) / p) := by |
calc
_ = ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * exp (-x)) := by
rw [← integral_comp_rpow_Ioi _ (one_div_ne_zero (ne_of_gt hp)),
abs_eq_self.mpr (le_of_lt (one_div_pos.mpr hp))]
refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)
rw [← rpow_mul (le_... | 2,359 |
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
open Real Set MeasureTheory MeasureTheory.Measure
section real
theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q +... | Mathlib/MeasureTheory/Integral/Gamma.lean | 39 | 57 | theorem integral_rpow_mul_exp_neg_mul_rpow {p q b : ℝ} (hp : 0 < p) (hq : - 1 < q) (hb : 0 < b) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- b * x ^ p) =
b ^ (-(q + 1) / p) * (1 / p) * Gamma ((q + 1) / p) := by |
calc
_ = ∫ x in Ioi (0:ℝ), b ^ (-p⁻¹ * q) * ((b ^ p⁻¹ * x) ^ q * rexp (-(b ^ p⁻¹ * x) ^ p)) := by
refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)
rw [mul_rpow _ (le_of_lt hx), mul_rpow _ (le_of_lt hx), ← rpow_mul, ← rpow_mul,
inv_mul_cancel, rpow_one, mul_assoc, ← mul_assoc, ← rpo... | 2,359 |
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
open Real Set MeasureTheory MeasureTheory.Measure
section real
theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q +... | Mathlib/MeasureTheory/Integral/Gamma.lean | 59 | 63 | theorem integral_exp_neg_rpow {p : ℝ} (hp : 0 < p) :
∫ x in Ioi (0:ℝ), exp (- x ^ p) = Gamma (1 / p + 1) := by |
convert (integral_rpow_mul_exp_neg_rpow hp neg_one_lt_zero) using 1
· simp_rw [rpow_zero, one_mul]
· rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp))]
| 2,359 |
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
open Real Set MeasureTheory MeasureTheory.Measure
section real
theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q +... | Mathlib/MeasureTheory/Integral/Gamma.lean | 65 | 69 | theorem integral_exp_neg_mul_rpow {p b : ℝ} (hp : 0 < p) (hb : 0 < b) :
∫ x in Ioi (0:ℝ), exp (- b * x ^ p) = b ^ (- 1 / p) * Gamma (1 / p + 1) := by |
convert (integral_rpow_mul_exp_neg_mul_rpow hp neg_one_lt_zero hb) using 1
· simp_rw [rpow_zero, one_mul]
· rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp)), mul_assoc]
| 2,359 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 105 | 105 | theorem areaForm_to_volumeForm (x y : E) : ω x y = o.volumeForm ![x, y] := by | simp [areaForm]
| 2,360 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 109 | 113 | theorem areaForm_apply_self (x : E) : ω x x = 0 := by |
rw [areaForm_to_volumeForm]
refine o.volumeForm.map_eq_zero_of_eq ![x, x] ?_ (?_ : (0 : Fin 2) ≠ 1)
· simp
· norm_num
| 2,360 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 116 | 121 | theorem areaForm_swap (x y : E) : ω x y = -ω y x := by |
simp only [areaForm_to_volumeForm]
convert o.volumeForm.map_swap ![y, x] (_ : (0 : Fin 2) ≠ 1)
· ext i
fin_cases i <;> rfl
· norm_num
| 2,360 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 125 | 127 | theorem areaForm_neg_orientation : (-o).areaForm = -o.areaForm := by |
ext x y
simp [areaForm_to_volumeForm]
| 2,360 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 142 | 143 | theorem abs_areaForm_le (x y : E) : |ω x y| ≤ ‖x‖ * ‖y‖ := by |
simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y]
| 2,360 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 146 | 147 | theorem areaForm_le (x y : E) : ω x y ≤ ‖x‖ * ‖y‖ := by |
simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.volumeForm_apply_le ![x, y]
| 2,360 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 150 | 158 | theorem abs_areaForm_of_orthogonal {x y : E} (h : ⟪x, y⟫ = 0) : |ω x y| = ‖x‖ * ‖y‖ := by |
rw [o.areaForm_to_volumeForm, o.abs_volumeForm_apply_of_pairwise_orthogonal]
· simp [Fin.prod_univ_succ]
intro i j hij
fin_cases i <;> fin_cases j
· simp_all
· simpa using h
· simpa [real_inner_comm] using h
· simp_all
| 2,360 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 161 | 168 | theorem areaForm_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
[hF : Fact (finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x y : F) :
(Orientation.map (Fin 2) φ.toLinearEquiv o).areaForm x y =
o.areaForm (φ.symm x) (φ.symm y) := by |
have : φ.symm ∘ ![x, y] = ![φ.symm x, φ.symm y] := by
ext i
fin_cases i <;> rfl
simp [areaForm_to_volumeForm, volumeForm_map, this]
| 2,360 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 172 | 180 | theorem areaForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x y : E) :
o.areaForm (φ x) (φ y) = o.areaForm x y := by |
convert o.areaForm_map φ (φ x) (φ y)
· symm
rwa [← o.map_eq_iff_det_pos φ.toLinearEquiv] at hφ
rw [@Fact.out (finrank ℝ E = 2), Fintype.card_fin]
· simp
· simp
| 2,360 |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 58 | 63 | theorem continuousAt_oangle {x : V × V} (hx1 : x.1 ≠ 0) (hx2 : x.2 ≠ 0) :
ContinuousAt (fun y : V × V => o.oangle y.1 y.2) x := by |
refine (Complex.continuousAt_arg_coe_angle ?_).comp ?_
· exact o.kahler_ne_zero hx1 hx2
exact ((continuous_ofReal.comp continuous_inner).add
((continuous_ofReal.comp o.areaForm'.continuous₂).mul continuous_const)).continuousAt
| 2,361 |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 68 | 68 | theorem oangle_zero_left (x : V) : o.oangle 0 x = 0 := by | simp [oangle]
| 2,361 |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 73 | 73 | theorem oangle_zero_right (x : V) : o.oangle x 0 = 0 := by | simp [oangle]
| 2,361 |
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