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Sketch given sets of complex numbers in the complex plane

For each of the following, make a sketch of all complex numbers z in the complex plane which satisfy the given expression.

  1. |z| < 1.
  2. z + \overline{z} = 1.
  3. z - \overline{z} = 1.
  4. |z - 1| = |z+1|.
  5. |z-i| = |z+i|.
  6. z + \overline{z} = |z|^2.

  1. The complex numbers z such that |z| < 1 are a solid disk of radius 1 centered at the origin:
    Complex1
  2. We have

        \begin{align*}  z + \overline{z} = 1 && \implies && x+ iy + x - iy &= 1 \\  && \implies && 2x &=1 \\  && \implies && x &= \frac{1}{2}. \end{align*}

    The plot in the complex plane is a vertical line at x = \frac{1}{2},
    Complex2

  3. We have

        \begin{align*}  z - \overline{z} = i && \implies && x + iy - x + iy &= i \\  && \implies && y &= \frac{1}{2}. \end{align*}

    The plot in the complex plane is a horizontal line at y = \frac{1}{2}.

  4. We have

        \begin{align*}  |z - 1| = |z + 1| && \implies && |x+iy -1| &= |x+iy + 1| \\  && \implies && |(x-1) + i y | &= |(x+1) + iy | \\  && \implies && \sqrt{(x-1)^2 + y^2} &= \sqrt{(x+1)^2 + y^2} \\  && \implies && x^2 - 2x + 1 + y^2 &= x^2 + 2x + 1 + y^2 \\  && \implies && 4x &= 0 \\  && \implies && x &= 0. \end{align*}

    The plot is a vertical line at x = 0.

  5. We have

        \begin{align*}  |z - i| = |z + i| && \implies && |x + (y-1)i| &= |x + (y+1)i| \\  && \implies && \sqrt{x^2 + y^2 - 2y + 1} &= \sqrt{x^2 + y^2 + 2y + 1} \\  && \implies && x^2 + y^2 - 2y + 1 &= x^2 + y^2 + 2y + 1 \\  && \implies && y &= 0. \end{align*}

    The plot is a horizontal line at y = 0.

  6. We have

        \begin{align*}  z + \overline{z} = |z|^2 && \implies && x + iy + x - iy &= x^2 + y^2 \\  && \implies && 2x &= x^2 + y^2 \\  && \implies && y = \pm \sqrt{x(x-2)}. \end{align*}

    The plot is a circle of radius one centered at (1,0),
    Complex3

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