Describe image under mapping $w=\frac{z-i}{z+i}$ of set $\{z=x+iy : x>0, y>0 \}$

Question

I'm new to complex analysis. In polar form, when I plug in a general $z=re^{i\theta}$ to $w$, I end up with $$w(z)=\dfrac{r^2-2ir\cos\theta-1}{r^2+2r\sin\theta+1}$$

I know since $x,y>0$, I have $0 < \theta < \frac{\pi}{2}$, but that's all I can tell.

Where do I go from here?

Answer 1

Rewrite $w=\frac{z-i}{z+i}$ as $z = i\frac{1+w}{1-w} =re^{i\theta}, \theta\in (0,\frac\pi2)$. Then $$e^{i2\theta} =\frac{z}{\bar z} = \frac{i\frac{1+w}{1-w}}{-i\frac{1+\bar w}{1-\bar w}} = -\frac{(1-\bar w)(1+w)}{(1+\bar w)(1-w)}\implies |w - i\tan\theta |^2 = \sec^2\theta$$ which represents a family of circles with center $i\tan\theta$ and radius $\sec\theta$. Thus, along with $\text{Im}(w)=\frac{-2r\cos\theta}{r^2+2r\sin\theta+1}<0$, the image is a half unit-disk, the shaded area below.

Answer 2

This is a bilinear or Mobius transformation. f(z) = $\frac{az+b}{cz+d}$. I'm a fan of Linear Geometry by Rafael Artzy, which covers this in detail. For example, f(z) = az+b has w = b/(1-a) as an invariant (fixed) point. It is a rotation through angle a about w. An inversion is N(z) = 1/z. A translation T(z) = z+b is labeled $T_b$. If $\delta = ad-bc$, and f = $\frac{-\delta}{c^2}$, then a dilatation is $D_f$ and the full Mobius transformation B = $T_{d/c} N D_f T_{a/c} $. For your example, $\delta = 2i$. Mobius transformations map the set of all straight lines and circles into the set of all straight lines and circles.

Answer 3

Why even bother with polar coordinates?

$$w = \frac{x +(y-1)i}{x + (y+1)i} \times \frac{x -(y+1)i}{x - (y+1)i} ~=~ \frac{(x^2 + 1 - y^2) - i(2x)}{x^2 + (y+1)^2}.\tag1$$

Note that by multiplying through by the complex conjugate of the denominator, I created a fraction whose denominator is a real number.

Addendum Per OP's request
Under the assumption that $(x + iy) = z = re^{i\theta}$ convert equation (1) into polar coordinates, involving $r$ and $\theta$.

This is extremely ugly. Frankly, the analysis in RobertTheTutor's answer is way over my head, so my approach would be to do things from scratch.

Since the original constraint is $0 < x,y$, you can assume that $0 < \theta < (\pi/2).$

Further, you have that $r^2 = |z|^2 = (x^2 + y^2)$, while $|w|^2 = \frac{(x^2 + 1 - y^2)^2 + 4x^2}{[x^2 + (y+1)^2]^2}.$

Assuming that you wish to express $w = se^{i\alpha}$, you will have that $s^2 = |w|^2$ and that $\alpha$ is the unique angle, within a modulus of $(2\pi)$ such that

$\cos(\alpha) = \frac{x^2 + 1 - y^2}{|t|}$ and $\sin(\alpha) = \frac{-2x}{|t|},~$ where $t^2 = (x^2 + 1 - y^2)^2 + (-2x)^2.$

Here, since $r^2 = (x^2 + y^2)$, and since $\cos(\theta) = \frac{x}{r}, \sin(\theta) = \frac{y}{r}$, the challenge would be to somehow:

first express $s$ in terms of $(r)$ and trig functions involving $(\theta)$ and then express $\cos(\alpha), \sin(\alpha)$ in terms of $s,r$ and trig functions involving $(\theta).$

My response is based on nothing more than my surviving Chapter 1 of "An Introduction To Complex Function Theory" (Bruce Palka). Palka does discuss mobius transformations, but the pertinent problems (involving mobius transformations) at the back of chapter 1 are structured very narrowly.

Palka does mention that he will discuss mobius transformations in later chapters, but it is a safe bet that this is no walk in the park. I question the intent of your instructor.

Do you really think that if $z = re^{i\theta}$, that (for example) either $(z - i)$ or $\frac{z-i}{z+i}$ would be so easily expressible in polar coordinates involving $r$ and $\theta$?

If there is an elegant way to do this, I'd certainly like it explained to me, in terms compatible with someone who has only survived chapter 1 of the book.

This addendum is nothing more than an elaboration that if an expression for $w$ is required in polar coordinates involving $r$ and $(\theta)$ then I am way out of my depth.

About the only thing that I can say for sure, is that based on equation (1) above, $\sin(\alpha) < 0$, and the sign of $\cos(\alpha)$ will obviously be based on the sign of the expression $(x^2 + 1 - y^2).$

A cursory examination of equation (1) above does not yield any obvious insights to me, re constraints on $|w|^2.$