I want to find $\arg(\frac{z-2}{z+2})= \arg(z-2)-\arg(z+2)$.

So what will this expression represent ? The difference of the angle the lines make with positive x axis?

  • $\begingroup$ please use mathjax, this is really hard to understand $\endgroup$ Dec 9, 2017 at 2:43

1 Answer 1


Take a look at the rightmost figure at the bottom (the leftmost figure will be used at the end of this answer).

Let us concentrate on the blue circles. Their common property : all of them pass through 2 fixed points on the x-axis that are $(-2,0)$ and $(2,0)$. Let us call them $A$ and $B$. Let us call $O$ the midpoint of $AB$ and $M$ the "generic point" $(x,y)$.

$$\arg\left(\dfrac{z+2}{z-2}\right) = \arg(z-(-2)) - \arg(z-2)=\text{angle}(\vec{MA},\vec{MB})$$

(as you have done).

Let us take a football (or soccer) comparison. Consider line AB as the goal. Let the upper plane ($y > 0$) be the playground. I am a player and I want to know what are the curves along which I have an equal opportunity to target the goal. For example, how are situated points from which I see the goal

  • under a right angle ? It is, the (half) blue circle with diameter $AB$.

  • under a $120°$ angle ? It is the upper arc of another blue circle., etc...,

In all cases, we get circles (or more properly : arcs of circles) by the classical "inscribed angle property".

In other words, any upperplane arc of a blue circle is the loci of points $M$ such that angle $(AMB)$ is fixed. The same by symmetry for lowerplane.

Dualy, red circles are the loci of points such that $$\tag{1}\left|\dfrac{z+2}{z-2}\right|=const.$$

Moreover, any red circle is orthogonal to any blue circle. The explanation comes from the fact that

$$\tag{2} z \to Z=\dfrac{z+2}{z-2},$$

considered as a transformation from $\mathbb{C}$ onto itself, is a "conformal map", mapping the left image (cobweb) onto the right image, with angle preservation (here right angles preservation). To be precise, any $z=re^{i\theta}$ on the left figure is mapped onto an image point characterized by $Z$ given by (1).

You might think that we are far away from your initial question ; in fact, if you understand this, you will see later on that we are on a central theme of complex function theory. In particular with homographic transforms like in (2).

Hope it helps your understanding.

enter image description here


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.