Is the Inverse Function Theorem Applicable to Complex Arcsin on C-{Pi/2+k*Pi}?

[RoNN|3]

It seems to me that the standard definition of the complex Arcsin (the principal branch) is something like this:
[PLAIN]http://math.fullerton.edu/mathews/c2003/maptrigonometricfun/MapTrigonometricFunMod/Images/mat1017.gif

Anyways, it's defined as a map from C-{z in R : |z|>=1} to the strip {z : -Pi/2<Re(z)<Pi/2} and I suppose that it can be extended to a map from C-{-1,1} to the "extended" strip {z : -Pi/2<Re(z)<Pi/2} U {z : Re(z)=-Pi/2, Im(z)>0} U {z : Re(z)=Pi/2, Im(z)<0}. This new strip can be translated by k*Pi to give a disjoint family that covers C-{Pi/2+k*Pi}.

Here's what I want to know. Isn't it true that the Inverse Function Theorem states that Sin is locally invertible on all of C-{Pi/2+k*Pi}? For points that lie in the open strips defined above this is ok, but what if the point z is such that Re(z)=Pi/2 for example? I need an open neighborhood of such z on which Arcsin makes sense (a neighborhood that contains points in both strips).

I hope it's clear what I want to say. If not, I will make some pictures to clarify.

 

[RoNN|3]

I seem to have overseen the fact that Sin(z+Pi)=-Sin(z) :)