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) => Complex.arg ↑z

@[simp]

theorem circle.arg_eq_arg {z : ↥circle} {w : ↥circle} : Complex.arg ↑z = Complex.arg ↑w ↔ z = w

theorem arg_expMapCircle {x : ℝ} (h₁ : -Real.pi < x) (h₂ : x ≤ Real.pi) : Complex.arg ↑(expMapCircle x) = x

@[simp]

theorem expMapCircle_arg (z : ↥circle) : expMapCircle (Complex.arg ↑z) = z

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

theorem circle.argPartialEquiv_symm_apply : ↑(PartialEquiv.symm circle.argPartialEquiv) = ⇑expMapCircle

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.argEquiv_symm_apply : ⇑circle.argEquiv.symm = ⇑expMapCircle ∘ Subtype.val

@[simp]

theorem circle.argEquiv_apply_coe (z : ↥circle) : ↑(circle.argEquiv z) = Complex.arg ↑z

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

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

Equations

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

theorem leftInverse_expMapCircle_arg : Function.LeftInverse (⇑expMapCircle) (Complex.arg ∘ Subtype.val)

theorem invOn_arg_expMapCircle : Set.InvOn (Complex.arg ∘ Subtype.val) (⇑expMapCircle) (Set.Ioc (-Real.pi) Real.pi) Set.univ

theorem surjOn_expMapCircle_neg_pi_pi : Set.SurjOn (⇑expMapCircle) (Set.Ioc (-Real.pi) Real.pi) Set.univ

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

theorem periodic_expMapCircle : Function.Periodic (⇑expMapCircle) (2 * Real.pi)

@[simp]

theorem expMapCircle_two_pi : expMapCircle (2 * Real.pi) = 1

theorem expMapCircle_sub_two_pi (x : ℝ) : expMapCircle (x - 2 * Real.pi) = expMapCircle x

theorem expMapCircle_add_two_pi (x : ℝ) : expMapCircle (x + 2 * Real.pi) = expMapCircle x

noncomputable def Real.Angle.expMapCircle (θ : Real.Angle) : ↥circle

expMapCircle, applied to a Real.Angle.

Equations

• Real.Angle.expMapCircle θ = Function.Periodic.lift periodic_expMapCircle θ

@[simp]

theorem Real.Angle.expMapCircle_coe (x : ℝ) : Real.Angle.expMapCircle ↑x = expMapCircle x

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

@[simp]

theorem Real.Angle.expMapCircle_zero : Real.Angle.expMapCircle 0 = 1

@[simp]

theorem Real.Angle.expMapCircle_neg (θ : Real.Angle) : Real.Angle.expMapCircle (-θ) = (Real.Angle.expMapCircle θ)⁻¹

@[simp]

theorem Real.Angle.expMapCircle_add (θ₁ : Real.Angle) (θ₂ : Real.Angle) : Real.Angle.expMapCircle (θ₁ + θ₂) = Real.Angle.expMapCircle θ₁ * Real.Angle.expMapCircle θ₂

@[simp]

theorem Real.Angle.arg_expMapCircle (θ : Real.Angle) : ↑(Complex.arg ↑(Real.Angle.expMapCircle θ)) = θ