# Mobius Transformations

• Nov 6th 2007, 01:38 PM
scorpio1
Mobius Transformations
Unsure if this is the right section but here goes:

By coordinte geometry, any line or circle is of the form Ax+By+C($\displaystyle x^2 + y^2$) = D, where A, B, C, D are real constants.

Set $\displaystyle \frac{1}{z}$ = u + iv.

Show that u = $\displaystyle \frac{x}{x^2 + y^2}$ and v = $\displaystyle \frac{-y}{x^2 + y^2}$.

Deduce that the linear fractional transformation sends a line or a circle to another such.

I'm really quite stumped by this...I don't see where the 1/z comes into all this...and just in general don't know how to start with this?

Any hints or help would be greatly appreciated.
• Nov 7th 2007, 12:32 PM
CaptainBlack
Quote:

Originally Posted by scorpio1
Unsure if this is the right section but here goes:

By coordinte geometry, any line or circle is of the form Ax+By+C($\displaystyle x^2 + y^2$) = D, where A, B, C, D are real constants.

Set $\displaystyle \frac{1}{z}$ = u + iv.

Show that u = $\displaystyle \frac{x}{x^2 + y^2}$ and v = $\displaystyle \frac{-y}{x^2 + y^2}$.

Deduce that the linear fractional transformation sends a line or a circle to another such.

I'm really quite stumped by this...I don't see where the 1/z comes into all this...and just in general don't know how to start with this?

Any hints or help would be greatly appreciated.

You are dealing with a circle (or line) in the complex plain, thus z=x+iy. now proceed.

RonL