Documentation

Mathlib.Analysis.Complex.UnitDisc.Basic

PoincarΓ© disc #

In this file we define Complex.UnitDisc to be the unit disc in the complex plane. We also introduce some basic operations on this disc.

The complex unit disc, denoted as 𝔻 withinin the Complex namespace

Equations

The complex unit disc, denoted as 𝔻 withinin the Complex namespace

Equations
@[simp]
theorem Complex.UnitDisc.coe_inj {z w : UnitDisc} :
↑z = ↑w ↔ z = w
@[deprecated Complex.UnitDisc.norm_lt_one (since := "2025-02-16")]

Alias of Complex.UnitDisc.norm_lt_one.

@[deprecated Complex.UnitDisc.norm_ne_one (since := "2025-02-16")]

Alias of Complex.UnitDisc.norm_ne_one.

@[simp]
theorem Complex.UnitDisc.coe_mul (z w : UnitDisc) :
↑(z * w) = ↑z * ↑w

A constructor that assumes β€–zβ€– < 1 instead of dist z 0 < 1 and returns an element of 𝔻 instead of β†₯Metric.ball (0 : β„‚) 1.

Equations
@[simp]
theorem Complex.UnitDisc.coe_mk (z : β„‚) (hz : β€–zβ€– < 1) :
↑(mk z hz) = z
@[simp]
theorem Complex.UnitDisc.mk_coe (z : UnitDisc) (hz : ‖↑zβ€– < 1 := β‹―) :
mk (↑z) hz = z
@[simp]
theorem Complex.UnitDisc.mk_neg (z : β„‚) (hz : β€–-zβ€– < 1) :
mk (-z) hz = -mk z β‹―
@[simp]
theorem Complex.UnitDisc.coe_zero :
↑0 = 0
@[simp]
theorem Complex.UnitDisc.coe_eq_zero {z : UnitDisc} :
↑z = 0 ↔ z = 0
@[simp]
theorem Complex.UnitDisc.mk_zero :
mk 0 β‹― = 0
@[simp]
theorem Complex.UnitDisc.mk_eq_zero {z : β„‚} (hz : β€–zβ€– < 1) :
mk z hz = 0 ↔ z = 0
@[simp]
theorem Complex.UnitDisc.coe_smul_circle (z : Circle) (w : UnitDisc) :
↑(z β€’ w) = ↑z * ↑w
@[simp]
theorem Complex.UnitDisc.coe_smul_closedBall (z : ↑(Metric.closedBall 0 1)) (w : UnitDisc) :
↑(z β€’ w) = ↑z * ↑w

Real part of a point of the unit disc.

Equations

Imaginary part of a point of the unit disc.

Equations
@[simp]
theorem Complex.UnitDisc.re_coe (z : UnitDisc) :
(↑z).re = z.re
@[simp]
theorem Complex.UnitDisc.im_coe (z : UnitDisc) :
(↑z).im = z.im
@[simp]
@[simp]

Conjugate point of the unit disc.

Equations
@[simp]
theorem Complex.UnitDisc.coe_conj (z : UnitDisc) :
↑z.conj = (starRingEnd β„‚) ↑z
@[simp]
@[simp]
@[simp]
theorem Complex.UnitDisc.conj_mul (z w : UnitDisc) :
(z * w).conj = z.conj * w.conj