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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?

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

1 Answer 1

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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

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