Question

Find a conformal mapping which maps the upper half-plane onto the exterior of the semi-infinite strip |Re w|< 1, Im w > 0

Answer 1

Let H be the half plane {z∈C | ℑz>0} and :

We have :

Thus, the preimage of H under cosh is an union of semi-infinite strips :

Let SS be the semi-infinite strip

First, let's show that cosh|S is injective. Let z,z′∈S such that We have

were

when then

then we get

which is not possible for

Then finally states that,Thus cosh|S is injective.

Thus the cosh being a holomorphic function, its image is open. cosh sends the boundary of S onto the boundary of H, and tends towards infinity as its argument grows inside S. Thus cosh|S is a proper function, which implies that its image is closed. Since H is connected, any subset of H which is non-empty, open and closed must equal H. Finally, cosh|S is also surjective,