Graphically showing $\operatorname{Re}\left(\frac{1+z}{1-z}\right) = 0$ for $z=\cos\theta+i\sin\theta\neq 1$

complex numberscomplex-analysispolar coordinates

The given complex number is $z = \cos\theta + i\sin\theta$, where $z$ is not $1$

I have to show that $$\operatorname{Re}\left(\frac{1+z}{1-z}\right) = 0$$

Algebraically, I have managed to do that using trigonometry and Euler's equation. But I can't succeed to imagine it on an Argand diagram using basic angle rules, without solving anything. In this case division, so angle subtraction. The angle should either be $90^\circ$ or $270^\circ$ since it should end up only on the complex part, I tried drawing it on the whiteboard but perhaps I am doing something wrong since I cannot get it to add up. Conceptually this is right.

Best Answer

One way could be to note that, not just this problem, but any linear fractional transformation $$z\mapsto\frac{az+b}{cz+d}$$ can always be seen as a composition of inversions, reflections, translations, and homotheties. You only need to reduce the fraction, to see the composition of these transformations.

$$\frac{1+z}{1-z}=1+\frac{2z}{1-z}=1+\frac{2}{1/z-1}$$

Therefore, to get the left hand side, we are composing $z\mapsto 1/z$, $z\mapsto z-1$, $z\mapsto 1/z$, $z\mapsto 2z$, $z\mapsto 1+z$, and $z\mapsto \operatorname{Re}(z)$

  • The map $z\mapsto 1/z$ is an inversion with respect to the unit circle, followed by a reflection with respect to the real axis.
  • The map $z\mapsto z-1$ is a translation one unit to the left.
  • The map $z\mapsto 2z$ is a homothety with center $0$ and ratio $2$.
  • The map $z\mapsto 1+z$ is a translation one unit to the right.
  • The map $z\mapsto\operatorname{Re}(z)$ is the orthogonal projection to the real axis.

If you only care about what happens to the unit circle, you can follow each transformation.

  • The inversion of the unit circle with respect to the unit circle is the identity. The reflection with respect to the real axis, well, flips the circle.
  • The translation one unit to the left you know.
  • Next there is another inversion with respect to the unit circle, but since now the circle passes through the origin, it becomes a straight line perpendicular to the real axis and passing through $-1/2$.
  • The homothety with ratio $2$, that will turn the line to another vertical line, but passing through $-1$.
  • There is a one unit translation to the right. This brings the vertical line to the imaginary axis.
  • Finally, the orthogonal projection to the real axis maps this vertical line to $0$.
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