Documentation

Mathlib.Analysis.Complex.UnitDisc.Basic

Poincaré 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.

Complex unit disc.

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    Complex unit disc.

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    Instances For
      theorem Complex.UnitDisc.abs_lt_one (z : Complex.UnitDisc) :
      Complex.abs z < 1
      theorem Complex.UnitDisc.abs_ne_one (z : Complex.UnitDisc) :
      Complex.abs z 1
      theorem Complex.UnitDisc.normSq_lt_one (z : Complex.UnitDisc) :
      Complex.normSq z < 1
      @[simp]
      theorem Complex.UnitDisc.coe_mul (z : Complex.UnitDisc) (w : Complex.UnitDisc) :
      (z * w) = z * w
      def Complex.UnitDisc.mk (z : ) (hz : Complex.abs z < 1) :

      A constructor that assumes abs z < 1 instead of dist z 0 < 1 and returns an element of 𝔻 instead of ↥Metric.ball (0 : ℂ) 1.

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      Instances For
        @[simp]
        theorem Complex.UnitDisc.coe_mk (z : ) (hz : Complex.abs z < 1) :
        @[simp]
        theorem Complex.UnitDisc.mk_coe (z : Complex.UnitDisc) (hz : optParam (Complex.abs z < 1) ) :
        @[simp]
        theorem Complex.UnitDisc.mk_neg (z : ) (hz : Complex.abs (-z) < 1) :
        Equations
        • One or more equations did not get rendered due to their size.
        @[simp]
        @[simp]
        @[simp]
        theorem Complex.UnitDisc.coe_smul_circle (z : circle) (w : Complex.UnitDisc) :
        (z w) = z * w
        @[simp]
        theorem Complex.UnitDisc.coe_smul_closedBall (z : (Metric.closedBall 0 1)) (w : Complex.UnitDisc) :
        (z w) = z * w

        Real part of a point of the unit disc.

        Equations
        • z.re = (z).re
        Instances For

          Imaginary part of a point of the unit disc.

          Equations
          • z.im = (z).im
          Instances For
            @[simp]
            theorem Complex.UnitDisc.re_coe (z : Complex.UnitDisc) :
            (z).re = z.re
            @[simp]
            theorem Complex.UnitDisc.im_coe (z : Complex.UnitDisc) :
            (z).im = z.im
            @[simp]
            theorem Complex.UnitDisc.re_neg (z : Complex.UnitDisc) :
            (-z).re = -z.re
            @[simp]
            theorem Complex.UnitDisc.im_neg (z : Complex.UnitDisc) :
            (-z).im = -z.im

            Conjugate point of the unit disc.

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              @[simp]
              @[simp]
              theorem Complex.UnitDisc.conj_conj (z : Complex.UnitDisc) :
              z.conj.conj = z
              @[simp]
              theorem Complex.UnitDisc.conj_neg (z : Complex.UnitDisc) :
              (-z).conj = -z.conj
              @[simp]
              theorem Complex.UnitDisc.re_conj (z : Complex.UnitDisc) :
              z.conj.re = z.re
              @[simp]
              theorem Complex.UnitDisc.im_conj (z : Complex.UnitDisc) :
              z.conj.im = -z.im
              @[simp]
              theorem Complex.UnitDisc.conj_mul (z : Complex.UnitDisc) (w : Complex.UnitDisc) :
              (z * w).conj = z.conj * w.conj