File: Mathlib/Analysis/SpecialFunctions/PolarCoord.lean

full_name: polarCoord_source
code: /-- The polar coordinates partial homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
@[simps]
def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) 
goal: polarCoord.source = {q | 0 < q.1} ∪ {q | q.2 ≠ 0}

full_name: polarCoord_target
code: /-- The polar coordinates partial homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
@[simps]
def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) 
goal: polarCoord.target = Set.Ioi 0 ×ˢ Set.Ioo (-Real.pi) Real.pi

full_name: polarCoord_apply
code: /-- The polar coordinates partial homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
@[simps]
def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) 
goal: ↑polarCoord q = (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)

full_name: polarCoord_symm_apply
code: /-- The polar coordinates partial homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
@[simps]
def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) 
goal: ↑polarCoord.symm p = (p.1 * Real.cos p.2, p.1 * Real.sin p.2)

full_name: polarCoord
code: /-- The polar coordinates partial homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. -/
@[simps]
def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) 
goal: PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)

full_name: hasFDerivAt_polarCoord_symm
code: 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 
goal: HasFDerivAt (↑polarCoord.symm)
  (LinearMap.toContinuousLinearMap
    ((Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ))
      !![Real.cos p.2, -p.1 * Real.sin p.2; Real.sin p.2, p.1 * Real.cos p.2]))
  p

full_name: instIsAddHaarMeasureProdRealVolume
code: instance : Measure.IsAddHaarMeasure volume (G := ℝ × ℝ) 
goal: MeasureTheory.volume.IsAddHaarMeasure

full_name: polarCoord_source_ae_eq_univ
code: theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ 
goal: polarCoord.source =ᵐ[MeasureTheory.volume] Set.univ

full_name: integral_comp_polarCoord_symm
code: theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (f : ℝ × ℝ → E) :
    (∫ p in polarCoord.target, p.1 • f (polarCoord.symm p)) = ∫ p, f p 
goal: ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑polarCoord.symm p) = ∫ (p : ℝ × ℝ), f p

full_name: Complex.polarCoord
code: /-- The polar coordinates partial homeomorphism in `ℂ`, mapping `r (cos θ + I * sin θ)` to `(r, θ)`.
It is a homeomorphism between `ℂ - ℝ≤0` and `(0, +∞) × (-π, π)`. -/
protected noncomputable def polarCoord : PartialHomeomorph ℂ (ℝ × ℝ) 
goal: PartialHomeomorph ℂ (ℝ × ℝ)

full_name: Complex.polarCoord_apply
code: protected theorem polarCoord_apply (a : ℂ) :
    Complex.polarCoord a = (Complex.abs a, Complex.arg a) 
goal: ↑Complex.polarCoord a = (Complex.abs a, a.arg)

full_name: Complex.polarCoord_source
code: protected theorem polarCoord_source : Complex.polarCoord.source = slitPlane 
goal: Complex.polarCoord.source = Complex.slitPlane

full_name: Complex.polarCoord_target
code: protected theorem polarCoord_target :
    Complex.polarCoord.target = Set.Ioi (0 : ℝ) ×ˢ Set.Ioo (-π) π 
goal: Complex.polarCoord.target = Set.Ioi 0 ×ˢ Set.Ioo (-Real.pi) Real.pi

full_name: Complex.polarCoord_symm_apply
code: @[simp]
protected theorem polarCoord_symm_apply (p : ℝ × ℝ) :
    Complex.polarCoord.symm p = p.1 * (Real.cos p.2 + Real.sin p.2 * Complex.I) 
goal: ↑Complex.polarCoord.symm p = ↑p.1 * (↑(Real.cos p.2) + ↑(Real.sin p.2) * Complex.I)

full_name: Complex.polarCoord_symm_abs
code: theorem polarCoord_symm_abs (p : ℝ × ℝ) :
    Complex.abs (Complex.polarCoord.symm p) = |p.1| 
goal: Complex.abs (↑Complex.polarCoord.symm p) = |p.1|

full_name: Complex.integral_comp_polarCoord_symm
code: protected theorem integral_comp_polarCoord_symm {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] (f : ℂ → E) :
    (∫ p in polarCoord.target, p.1 • f (Complex.polarCoord.symm p)) = ∫ p, f p 
goal: ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑Complex.polarCoord.symm p) = ∫ (p : ℂ), f p