Conformal Mapping: Sketch Regions & Find Mapping

[diorific]

Hi, I need to sketche ach of the following regions: R = {z :|z| < √2, 7π/16 < Argz<9π/16}, R1 = {z :|z| < 16, Rez>0} and write down a one-one conformal mapping f1 from R onto R1.

Here is my sketch https://onedrive.live.com/redir?resid=4cdf33ffa97631ef%2110238

But I'm finding hard to find the mapping. So if someone can help me out that'd be appreciated.

 

[The_Duck]

As a start, I'd consider the mappings $z \to z^\alpha$ for various powers $\alpha$ and see what effect they have on $R$.

 

[diorific]

Ok thank you I think it's z→z^8