Trouble finding a mobius transformation from a domain to a unit disc

[librastar]

Homework Statement

S = { z | |Im(z)| < 5 }, z is a complex number

Homework Equations

I am trying to generate a mobius transformation w = f(z) such that it will map S onto a unit disc but I keep running into problems and contradictions. I think there is a big mistake in my attempt but I can't figure it out, can somebody please give me a hint on what is wrong?

The Attempt at a Solution

First I know that |Im(z)| < 5 implies that -5 < y < 5 and I want to find a mobius transformation such that S will be mapped onto unit disc, | z | < or = 1.
I start with the standard Mobius transformation form:

f(z) = (az+b)/(cz+d), assume c doesn't equal 0, and I want the following,
0 -> 0
5i -> i
infinity -> 1

Then f(0) = b/d = 0, I choose b = 0 instead of d = infinity.
f(infinity) = a/c = 1, this shows a = c
f(5i) = (5ai)/(5ci + d) = (-25ac - 5adi)/((-25c^2) - d^2) = (25ac + 5adi)/(25c^2 + d^2)
However at this point I figure out that (25c^2)/(25c^2 + d^2) = 0, this implies that c = 0.
Since c = 0 and a = c, then a = 0.
Now I can't figure out what d is, since I have 0/(d^2) = 1, which makes no sense.
In addition because a = b = c = 0, ad - bc = 0 which violates the condition of Mobius transformation.

 

[Tinyboss]

First I know that Im(z) < 5 implies that -5 < y < 5 and...

There's your first problem.

 

[librastar]

There's your first problem.

Sorry I forgot to add the mod, it should be |Im(z)|<5

 

[Tinyboss]

Mobius transformations take generalized circles to generalized circles. The boundary of the unit disk is a circle. What's the boundary of the set you're starting with? Is it a (generalized) circle?

 

[librastar]

Mobius transformations take generalized circles to generalized circles. The boundary of the unit disk is a circle. What's the boundary of the set you're starting with? Is it a (generalized) circle?

In this case, I think my S is not a generalized circle so I can't generate a mobius transformation?

If not, can I still find an analytic functions that maps S onto unit disc?