Can Mobius Transformations Map Circles Onto Circles?

[sara_87]

Homework Statement

I know that the Mobius transformation:

$$g(z) = \frac{z-z_1}{z-z_3}\frac{z_2-z_3}{z_2-z_1}$$

maps a circle (with points z_1, z_2, z_3 somewhere on the circumference) onto a line.

But, i want a general formula for f(z) that maps a circle (z_1,z_2,z_3) ontp a circle (w_1,w_2,w_3)

Homework Equations

The Attempt at a Solution

Does anyone have any ideas?

Thank you in advance

 

[JSuarez]

The inverse of g(z) maps a line into a circle; find the general expression for its inverse, select three points in a convinient line and compose the two.

 

[sara_87]

I found the inverse. It's z= a function in terms of w and z_1, z_2, z_3.

Why does this help? and what do you mean 'compose the two' ?

 

[JSuarez]

g(z) takes the points (z1,z2,z3) of the circle to (g(z1),g(z2),g(z3)) in a line. Then choose the inverse f(z) such that it takes the points (g(z1),g(z2),g(z3)) in the line, to (w1,w2,w3) in the circle. The composition f(g(z)) will take (z1,z2,z3) in (w1,w2,w3).

 

[sara_87]

the inverse f(z) such that it takes the points (g(z1),g(z2),g(z3)) in the line, to (w1,w2,w3) in the circle is:

$$f(g(z))=\frac{w-w_1}{w-w_3}\frac{w_2-w_3}{w_2-w_1}$$

so now i have to combine this with the g(z) is gave in the first post?
Why?