find a conformal map which maps a strip with a slit to the same strip without slit 
find a conformal map which maps $\left\{ z: 0<\operatorname{Im z}<1\right\}$ minus $[a,a+hi]$ into the same strip without slit, where $a\in \mathbb{R}$ and $0<h<1$.

since the problem asks to eliminate slit I want to do $z^2, \sqrt{z}$ transformation to make it into a non-slit region (like first move the slit to the imaginary axis, then do exp, move it to the negative real axis, take square root to get a half plane) The problem is I don't know how to map a plane/half plane/etc. into a strip again, and I found no problem discussing on this kind of map. Should I try to inverse the map of a strip to something or what? Thanks for any help.

If you could find a map which maps $\{z:0<\operatorname{Im} z<1\}\text{ minus }[a,a+hi]$ onto the upper half plane, then consider $$
z\mapsto \frac{1}{\pi}\log z,$$
which maps the upper half plane onto $\{z:0<\operatorname{Im} z<1\}$.
