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## Homework Statement

For the mapping f(z) = [itex]\frac{z - i}{z + i}[/itex], find the image of Im(z) ≥ 0

## The Attempt at a Solution

So in the z-plane this is clearly everything for y ≥ 0

I substituted x + iy, and rationalised it to get:

[itex]u+iv[/itex] = [itex]{\frac {{x}^{2}-1+{y}^{2}}{{x}^{2}+1+{y}^{2}+2\,y}}-{\frac {2\,ix}{{x}

^{2}+1+{y}^{2}+2\,y}}

[/itex]

The denominator can be simplified to x^2 + (y+1)^2

The answer says that this is a closed disc of |w| ≤ 1 with an exclusion at w = 1. Which means I should be seeing an answer of the form u^2 + v^2 ≤ 1

I do not see how they got this.

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