# Complex analysis question

1. Oct 1, 2013

### Genericcoder

1. The problem statement, all variables and given/known data

Let S = {z : 1<= Im(z) <=2}. Determine f(S) if f: S ->C
defined by
f(z) = (z + 1) / (z - 1)

2. Relevant equations

z = x + iy

3. The attempt at a solution
[attempt at solution]

so here my solution

f(z) = 1 + 2/(z - 1)

after doing some algebra <-> f(z) = x^2 + y^2/((x - 1)^2 + y^2) - [2y/((x-1)^2 + y^2)]i

therefore Im(z) = -2y/((x - 1)^2 + y^2) so F(S) = {z : 1<= (-2y)/((x-1)^2 + y^2) <= 2}
but I am stuck at this point I don't know wat does this represent.

2. Oct 1, 2013

### Dick

Concentrate on what the boundaries of your region are. For example, if 1=(-2y)/((x-1)^2 + y^2) what kind of curve is that? Multiply it out and complete the square. At a more abstract level f(z) is a Mobius transformation. It will map lines to lines or circles, yes?

Last edited: Oct 1, 2013
3. Oct 2, 2013

### Genericcoder

yes I did that I got something weird

I got (x-1)^2 + y^2 <= -2y <= 2( (x - 1)^2 + y^2)) the way I see it its between two circles but how to show that ???

4. Oct 2, 2013

### Dick

Just look at the boundaries. Where your inequality becomes an equality. 1=(-2y)/((x-1)^2 + y^2) and 2=(-2y)/((x-1)^2 + y^2). What are the boundary curves? And yes, they are two circles.

5. Oct 2, 2013

### Genericcoder

o I see I figured it out ty alot Dick!