Maps on the unit circle

In this file we prove some basic lemmas about expMapCircle and the restriction of Complex.arg to the unit circle. These two maps define a partial equivalence between circle and ℝ, see circle.argPartialEquiv and circle.argEquiv, that sends the whole circle to (-π, π].

theorem Circle.injective_arg : Function.Injective fun (z : Circle) => (↑z).arg

@[simp]

theorem Circle.arg_eq_arg {z w : Circle} : (↑z).arg = (↑w).arg ↔ z = w

theorem Circle.arg_exp {x : ℝ} (h₁ : -Real.pi < x) (h₂ : x ≤ Real.pi) : (↑(exp x)).arg = x

@[simp]

theorem Circle.exp_arg (z : Circle) : exp (↑z).arg = z

noncomputable def Circle.argPartialEquiv : PartialEquiv Circle ℝ

Complex.arg ∘ (↑) and expMapCircle define a partial equivalence between circle and ℝ with source = Set.univ and target = Set.Ioc (-π) π.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Circle.argPartialEquiv_source : argPartialEquiv.source = Set.univ

@[simp]

theorem Circle.argPartialEquiv_target : argPartialEquiv.target = Set.Ioc (-Real.pi) Real.pi

@[simp]

theorem Circle.argPartialEquiv_symm_apply : ↑argPartialEquiv.symm = ⇑exp

@[simp]

theorem Circle.argPartialEquiv_apply : ↑argPartialEquiv = Complex.arg ∘ Subtype.val

noncomputable def Circle.argEquiv : Circle ≃ ↑(Set.Ioc (-Real.pi) Real.pi)

Complex.arg and expMapCircle define an equivalence between circle and (-π, π].

Equations

• Circle.argEquiv = { toFun := fun (z : Circle) => ⟨(↑z).arg, ⋯⟩, invFun := ⇑Circle.exp ∘ Subtype.val, left_inv := ⋯, right_inv := Circle.argEquiv._proof_3 }

@[simp]

theorem Circle.argEquiv_apply_coe (z : Circle) : ↑(argEquiv z) = (↑z).arg

@[simp]

theorem Circle.argEquiv_symm_apply : ⇑argEquiv.symm = ⇑exp ∘ Subtype.val

theorem Circle.leftInverse_exp_arg : Function.LeftInverse (⇑exp) (Complex.arg ∘ Subtype.val)

theorem Circle.invOn_arg_exp : Set.InvOn (Complex.arg ∘ Subtype.val) (⇑exp) (Set.Ioc (-Real.pi) Real.pi) Set.univ

theorem Circle.surjOn_exp_neg_pi_pi : Set.SurjOn (⇑exp) (Set.Ioc (-Real.pi) Real.pi) Set.univ

theorem Circle.exp_eq_exp {x y : ℝ} : exp x = exp y ↔ ∃ (m : ℤ), x = y + ↑m * (2 * Real.pi)

theorem Circle.periodic_exp : Function.Periodic (⇑exp) (2 * Real.pi)

@[simp]

theorem Circle.exp_two_pi : exp (2 * Real.pi) = 1

theorem Circle.exp_int_mul_two_pi (n : ℤ) : exp (↑n * (2 * Real.pi)) = 1

theorem Circle.exp_two_pi_mul_int (n : ℤ) : exp (2 * Real.pi * ↑n) = 1

theorem Circle.exp_eq_one {r : ℝ} : exp r = 1 ↔ ∃ (n : ℤ), r = ↑n * (2 * Real.pi)

theorem Circle.exp_inj {r s : ℝ} : exp r = exp s ↔ r ≡ s [PMOD 2 * Real.pi]

theorem Circle.exp_sub_two_pi (x : ℝ) : exp (x - 2 * Real.pi) = exp x

theorem Circle.exp_add_two_pi (x : ℝ) : exp (x + 2 * Real.pi) = exp x

noncomputable def Real.Angle.toCircle (θ : Angle) : Circle

Circle.exp, applied to a Real.Angle.

Equations

• θ.toCircle = Circle.periodic_exp.lift θ

@[simp]

theorem Real.Angle.toCircle_coe (x : ℝ) : (↑x).toCircle = Circle.exp x

theorem Real.Angle.coe_toCircle (θ : Angle) : ↑θ.toCircle = ↑θ.cos + ↑θ.sin * Complex.I

@[simp]

theorem Real.Angle.toCircle_zero : toCircle 0 = 1

@[simp]

theorem Real.Angle.toCircle_neg (θ : Angle) : (-θ).toCircle = θ.toCircle⁻¹

@[simp]

theorem Real.Angle.toCircle_add (θ₁ θ₂ : Angle) : (θ₁ + θ₂).toCircle = θ₁.toCircle * θ₂.toCircle

@[simp]

theorem Real.Angle.arg_toCircle (θ : Angle) : ↑(↑θ.toCircle).arg = θ

Map from AddCircle to Circle

theorem AddCircle.scaled_exp_map_periodic {T : ℝ} : Function.Periodic (fun (x : ℝ) => Circle.exp (2 * Real.pi / T * x)) T

noncomputable def AddCircle.toCircle {T : ℝ} : AddCircle T → Circle

The canonical map fun x => exp (2 π i x / T) from ℝ / ℤ • T to the unit circle in ℂ. If T = 0 we understand this as the constant function 1.

Equations

• AddCircle.toCircle = ⋯.lift

theorem AddCircle.toCircle_apply_mk {T : ℝ} (x : ℝ) : toCircle ↑x = Circle.exp (2 * Real.pi / T * x)

theorem AddCircle.toCircle_add {T : ℝ} (x y : AddCircle T) : (x + y).toCircle = x.toCircle * y.toCircle

@[simp]

theorem AddCircle.toCircle_zero {T : ℝ} : toCircle 0 = 1

theorem AddCircle.continuous_toCircle {T : ℝ} : Continuous toCircle

theorem AddCircle.injective_toCircle {T : ℝ} (hT : T ≠ 0) : Function.Injective toCircle

noncomputable def AddCircle.homeomorphCircle' : AddCircle (2 * Real.pi) ≃ₜ Circle

The homeomorphism between AddCircle (2 * π) and Circle.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AddCircle.homeomorphCircle'_symm_apply (x : Circle) : homeomorphCircle'.symm x = ↑(↑x).arg

@[simp]

theorem AddCircle.homeomorphCircle'_apply (θ : Real.Angle) : homeomorphCircle' θ = θ.toCircle

theorem AddCircle.homeomorphCircle'_apply_mk (x : ℝ) : homeomorphCircle' ↑x = Circle.exp x

noncomputable def AddCircle.homeomorphCircle {T : ℝ} (hT : T ≠ 0) : AddCircle T ≃ₜ Circle

The homeomorphism between AddCircle and Circle.

Equations

• AddCircle.homeomorphCircle hT = (AddCircle.homeomorphAddCircle T (2 * Real.pi) hT AddCircle.homeomorphCircle._proof_1).trans AddCircle.homeomorphCircle'

theorem AddCircle.homeomorphCircle_apply {T : ℝ} (hT : T ≠ 0) (x : AddCircle T) : (homeomorphCircle hT) x = x.toCircle

theorem isLocalHomeomorph_circleExp : IsLocalHomeomorph ⇑Circle.exp