mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hf : MellinConvergent f ↑σ
hFf : VerticalIntegrable (mellin f) σ volume
hfx : ContinuousAt f x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
⊢ Integrable g volume

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hFf : VerticalIntegrable (mellin f) σ volume
hfx : ContinuousAt f x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
⊢ 2 * π ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: f : ℝ → ℂ
s a : ℂ
⊢ mellin (fun t ↦ f t / a) s = mellin f s / a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hFf : VerticalIntegrable (mellin f) σ volume
hfx : ContinuousAt f x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
h2π : 2 * π ≠ 0
⊢ Integrable (𝓕 fun u ↦ rexp (-(σ * u)) • f (rexp (-u))) volume

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hFf : VerticalIntegrable (mellin f) σ volume
hfx : ContinuousAt f x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
⊢ Integrable (𝓕 g) volume

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hfx : ContinuousAt f x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
⊢ ContinuousAt (fun u ↦ rexp (-σ * u)) (-Real.log x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hfx : ContinuousAt f x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
⊢ ContinuousAt (fun u ↦ rexp (-u)) (-Real.log x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hfx : ContinuousAt f x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
⊢ ContinuousAt g (-Real.log x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ mellinInv σ (mellin f) x = mellinInv σ (fun s ↦ 𝓕 g (s.im / (2 * π))) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝³¹ : NormedField 𝕜₁✝
inst✝³⁰ : NormedField 𝕜₂✝
σ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
inst✝²⁹ : AddCommGroup E✝
inst✝²⁸ : TopologicalSpace E✝
inst✝²⁷ : Module 𝕜₁✝ E✝
inst✝²⁶ : AddCommGroup F✝
inst✝²⁵ : TopologicalSpace F✝
inst✝²⁴ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
𝕜₄ : Type u_8
E : Type u_9
F : Type u_10
G : Type u_11
H : Type u_12
inst✝²³ : NormedField 𝕜₁
inst✝²² : NormedField 𝕜₂
inst✝²¹ : NormedField 𝕜₃
inst✝²⁰ : NormedField 𝕜₄
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₁₄ : 𝕜₁ →+* 𝕜₄
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₃₄ : 𝕜₃ →+* 𝕜₄
inst✝¹⁹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁸ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
inst✝¹⁶ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup F
inst✝¹¹ : TopologicalSpace F
inst✝¹⁰ : Module 𝕜₂ F
inst✝⁹ : AddCommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : AddCommGroup H
inst✝⁵ : TopologicalSpace H
inst✝⁴ : Module 𝕜₄ H
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
x✝ : F →WOT[𝕜₂] F
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] fun x ↦ x✝ * x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ mellinInv σ (fun s ↦ 𝓕 g (s.im / (2 * π))) x = ↑x ^ (-↑σ) • g (-Real.log x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ ↑x ^ (-↑σ) • g (-Real.log x) = ↑x ^ (-↑σ) • rexp (σ * Real.log x) • f (rexp (Real.log x))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ ↑x ^ (-↑σ) • rexp (σ * Real.log x) • f (rexp (Real.log x)) = f x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hf : MellinConvergent f ↑σ
hFf : VerticalIntegrable (mellin f) σ volume
hfx : ContinuousAt f x
⊢ mellinInv σ (mellin f) x = f x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
⊢ mellin (fun t ↦ f (t ^ a)) s = |a|⁻¹ • mellin f (s / ↑a)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ 1 - s = -(s - 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝³¹ : NormedField 𝕜₁✝
inst✝³⁰ : NormedField 𝕜₂✝
σ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
inst✝²⁹ : AddCommGroup E✝
inst✝²⁸ : TopologicalSpace E✝
inst✝²⁷ : Module 𝕜₁✝ E✝
inst✝²⁶ : AddCommGroup F✝
inst✝²⁵ : TopologicalSpace F✝
inst✝²⁴ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
𝕜₄ : Type u_8
E : Type u_9
F : Type u_10
G : Type u_11
H : Type u_12
inst✝²³ : NormedField 𝕜₁
inst✝²² : NormedField 𝕜₂
inst✝²¹ : NormedField 𝕜₃
inst✝²⁰ : NormedField 𝕜₄
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₁₄ : 𝕜₁ →+* 𝕜₄
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₃₄ : 𝕜₃ →+* 𝕜₄
inst✝¹⁹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁸ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
inst✝¹⁶ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup F
inst✝¹¹ : TopologicalSpace F
inst✝¹⁰ : Module 𝕜₂ F
inst✝⁹ : AddCommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : AddCommGroup H
inst✝⁵ : TopologicalSpace H
inst✝⁴ : Module 𝕜₄ H
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
x✝ : F →WOT[𝕜₂] F
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] fun x ↦ x * x✝

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ (fun t ↦ ↑t ^ (s - 1) • f (a * t)) t = (fun t ↦ ↑a ^ (1 - s) • (fun u ↦ ↑u ^ (s - 1) • f u) (a * t)) t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : 1 + z ∈ slitPlane
⊢ ContinuousOn (fun t ↦ 1 + t • z) (Set.Icc 0 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : 1 + z ∈ slitPlane
⊢ log (1 + z) = z * ∫ (t : ℝ) in 0..1, (1 + t • z)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : 1 - z ∈ slitPlane
⊢ log (1 - z)⁻¹ = z * ∫ (t : ℝ) in 0..1, (1 - t • z)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: ⊢ logTaylor 0 = fun x ↦ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
⊢ logTaylor (n + 1) = logTaylor n + fun z ↦ (-1) ^ (n + 1) * z ^ n / ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
⊢ mellin (fun t ↦ f (a * t)) s = ↑a ^ (-s) • mellin f s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
⊢ logTaylor n 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
⊢ mellin (fun t ↦ f (t * a)) s = ↑a ^ (-s) • mellin f s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
⊢ mellin (fun t ↦ f t⁻¹) s = mellin f (-s)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℝ → E
s : ℂ
hf : MellinConvergent f s
hg : MellinConvergent g s
⊢ MellinConvergent (fun t ↦ f t + g t) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
⊢ s.GammaIntegral = mellin (fun x ↦ ↑(rexp (-x))) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℝ → E
s : ℂ
hf : MellinConvergent f s
hg : MellinConvergent g s
⊢ mellin (fun t ↦ f t + g t) s = mellin f s + mellin g s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℝ → E
s : ℂ
hf : MellinConvergent f s
hg : MellinConvergent g s
⊢ MellinConvergent (fun t ↦ f t - g t) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℝ → E
s : ℂ
hf : MellinConvergent f s
hg : MellinConvergent g s
⊢ mellin (fun t ↦ f t - g t) s = mellin f s - mellin g s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
hf : MellinConvergent f s
R : Type u_2
inst✝³ : NormedRing R
inst✝² : Module R E
inst✝¹ : IsBoundedSMul R E
inst✝ : SMulCommClass ℂ R E
c : R
⊢ mellin (fun t ↦ c • f t) s = c • mellin f s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
n : ℕ
ih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z
⊢ HasDerivAt (fun x ↦ x ^ (n + 1) / (↑n + 1)) (z ^ n) z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : 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
⊢ AEStronglyMeasurable (fun t ↦ ↑t ^ (s - 1) • f t) (volume.restrict T)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
hs : 0 < s.re
⊢ 1 = ↑(rexp (-0))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
⊢ HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : 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 ↦ ↑t ^ (s - 1) • f t) T volume ↔ IntegrableOn (fun t ↦ t ^ (s.re - 1) * ‖f t‖) T volume

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
hs : 0 < s.re
⊢ HasDerivAt GammaIntegral (∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) * (↑(Real.log t) * ↑(rexp (-t)))) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ (n : ℕ) (s : ℂ), -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s
⊢ -↑(⌊-s.re⌋₊ + 1) < s.re

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : 1 + z ∈ slitPlane
⊢ -z ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
s : ℂ
hsre : -↑0 < s.re
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ 0 < s.re

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
hsre : 0 < s.re
this : IsOpen {s | 0 < s.re}
⊢ 0 < s.re

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
n : ℕ
IH : ∀ (s : ℂ), -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s
s : ℂ
hsre : -↑(n + 1) < s.re
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ s ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : 1 + z ∈ slitPlane
⊢ HasDerivAt (fun z ↦ log (1 + z) - logTaylor (n + 1) z) ((-z) ^ n * (1 + z)⁻¹) z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: t : ℝ
ht : 0 ≤ t ∧ t ≤ 1
z : ℂ
hz : ‖z‖ < 1
⊢ 0 < 1 - ‖z‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: t : ℝ
ht : 0 ≤ t ∧ t ≤ 1
z : ℂ
hz : ‖z‖ < 1
⊢ 1 - ‖z‖ ≤ 1 - t * ‖z‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
n : ℕ
IH : ∀ (s : ℂ), -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s
s : ℂ
hsre : -↑(n + 1) < s.re
hs : ∀ (m : ℕ), s ≠ -↑m
hsne : s ≠ 0
⊢ -↑n < (s + 1).re

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: t : ℝ
ht : 0 ≤ t ∧ t ≤ 1
z : ℂ
hz : ‖z‖ < 1
⊢ 1 - t * ‖z‖ = 1 - ‖↑t * z‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: t : ℝ
ht : 0 ≤ t ∧ t ≤ 1
z : ℂ
hz : ‖z‖ < 1
⊢ 1 - ‖↑t * z‖ ≤ ‖1 + ↑t * z‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: t : ℝ
ht : t ∈ Set.Icc 0 1
z : ℂ
hz : ‖z‖ < 1
⊢ ‖(1 + ↑t * z)⁻¹‖ ≤ (1 - ‖z‖)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
n : ℕ
IH : ∀ (s : ℂ), -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s
s : ℂ
hsre : -↑(n + 1) < s.re
hs : ∀ (m : ℕ), s ≠ -↑m
hsne : s ≠ 0
m : ℕ
⊢ s + 1 ≠ -↑m

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
this : ContinuousOn (fun t ↦ (1 + t • z)⁻¹) (Set.uIcc 0 1)
⊢ ContinuousOn (fun t ↦ t ^ n * ‖(1 + ↑t * z)⁻¹‖) (Set.uIcc 0 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
⊢ IntervalIntegrable (fun t ↦ t ^ n * ‖(1 + ↑t * z)⁻¹‖) MeasureTheory.volume 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
⊢ Continuous fun t ↦ t ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
n : ℕ
s : ℂ
IH : DifferentiableAt ℂ Gamma (s + 1)
hsre : -↑(n + 1) < s.re
hs : ∀ (m : ℕ), s ≠ -↑m
hsne : s ≠ 0
this : DifferentiableAt ℂ (Gamma ∘ fun s ↦ s + 1) s
⊢ ∀ (a : ℂ), a ≠ 0 → Gamma a = (Gamma ∘ fun s ↦ s + 1) a / id a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < max e 1
⊢ ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi (max e 1)), ‖t ^ (s - 1) * f t‖ ≤ t ^ (s - 1 + -a) * d

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
⊢ ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ Gamma s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: ⊢ ∀ (m : ℕ), 1 ≠ -↑m

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
⊢ Continuous fun t ↦ (-(↑t * z)) ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
⊢ ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: ⊢ ContinuousAt Gamma (0 + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: ⊢ Tendsto (fun z ↦ z * Gamma z) (𝓝[≠] 0) (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: ⊢ Disjoint (𝓝 1) (𝓝 (0 * Gamma 0))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < max e 1
this : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi (max e 1)), ‖t ^ (s - 1) * f t‖ ≤ t ^ (s - 1 + -a) * d
⊢ s - 1 + -a < -1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: n : ℕ
ih : ContinuousAt Gamma (-(↑n + 1))
this : ContinuousAt (fun s ↦ Gamma (s - 1 + 1)) (-↑n)
⊢ ContinuousAt Gamma (-↑n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
⊢ ∃ c, 0 < c ∧ IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioi c) volume

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
⊢ f z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: n : ℕ
⊢ ¬ContinuousAt Gamma (-↑n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
hs : s ≠ 0
h : ∀ (m : ℕ), s ≠ -↑m
this : HasDerivWithinAt (fun s ↦ Gamma (s + 1)) (Gamma s + s * deriv Gamma s) {0}ᶜ s
⊢ deriv Gamma (s + 1) = Gamma s + s * deriv Gamma s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
hz : z ≠ 0
x : ℝ
⊢ HasDerivAt (fun y ↦ Complex.sin (2 * z * ↑y) / (2 * z)) (Complex.cos (2 * z * ↑x)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
hs : s ≠ 0
⊢ deriv Gamma (s + 1) = Gamma s + s * deriv Gamma s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℝ Gamma s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: x✝ : ℝ
h : x✝ ∈ Ioi 0
⊢ ∀ (m : ℕ), x✝ ≠ -↑m

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
hz : z ≠ 0
x : ℝ
⊢ HasDerivAt (fun y ↦ -Complex.cos (2 * z * ↑y) / (2 * z)) (Complex.sin (2 * z * ↑x)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
⊢ b = 1 - a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
⊢ 0 < a * s + b * t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
a✝ : x ∈ uIcc 0 (π / 2)
⊢ HasDerivAt (fun y ↦ ↑(cos y)) (-↑(sin x)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: b : ℝ
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
hf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
s : ℝ
hs : b < s
d : ℝ
left✝ : d > 0
ε : ℝ
hε : 0 < ε
hε' : ∀ ⦃y : ℝ⦄, dist y 0 < ε → y ∈ Ioi 0 → ‖f y‖ ≤ d * ‖y ^ (-b)‖
⊢ -1 < s - b - 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
⊢ ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x

Tactic:
by

Goal state after: