Conformal mapping problem(bilinear type)

[poweng]

Homework Statement

Hi all. We are asked to transform the shaded area in below figure to between two concentric circles, an annulus. Where these circles' center will be is not important, just transform the area to between any two concentric circles. As you see in figure, shaded area is whole right half plane except a gap having center at x=a and radius b.

Figure is here.
http://img189.imageshack.us/img189/6331/conform.gif
You may suggest $$\frac{z+\alpha}{z+\beta}$$ type transformations which may map to origin centered circles.
My suggestion is w=$$\frac{1}{z+a}$$, no b, looks simple? I think this formula maps the shaded area to between circles centered at $$\frac{1}{2a}$$.
My explanation is: Shift right whole plane "a" much. Take $$\frac{1}{z}$$. If the gap were not there, taking $$\frac{1}{z}$$ yields a circle having center at$$\frac{1}{2a}$$ and being tangential to origin. The gap is there and it causes a little gap inside the former circle having center at $$\frac{1}{2a}$$. So, $$\frac{1}{z+a}$$ ends up with two concentric circles with transformed are between them. Am I wrong? My instructor denies me without making a reasonable explanation.

Supporting or refuting explanations will be appreciated. Thanks in advance.

 

[poweng]

I have noticed my mistake. Idea is correct but there is a little mistake. Correct transform is: 1/(z+sqrt(a^2-b^2)). Anybody, any comment?