# Complex analysis: mapping a hyperbola onto a line

## Homework Statement

We want to create a map from (x,y) to (u,v) such that the right side (positive x) of the hyperbola x^2 - y^2 = 1 is mapped onto the line v = 0 AND all the points to the left of that hyperbola are mapped to above the line. The mapping should be one-to-one and conformal

## Homework Equations

I think (but am not 100% sure) the map has to be from z = x+iy to w = u+iv, so it's important to define x and y in terms of z. I figured out that x = (z+conj(z))/2 and y = (z-conj(z))/2

## The Attempt at a Solution

We can map y directly to u through u = y. That should be sufficient I think.
v = x^2 - y^2 - 1 does map the parabola onto the line (it makes v = 0) but it maps the points to the right of the parabola onto above the line, so we multiply by -1 to get v = y^2 - x^2 + 1. We can write everything as:

w = y + i(y^2 - x^2 + 1) and then substitute the x = (z+conj(z))/2 and y = (z-conj(z))/2 in to make it a function of z.

This seems to work for when x >= 0, but it fails to map some points on the left side of the y axis (all of which should be mapped to above the line v = 0). This is because the function x^2 - y^2 = 1 describes two parabolas, and we have to somehow mathematically disregard the left one. I'm not sure how to do that. Could somebody please help?