Conformal mapping between two half space

[radiofeda]

Hi all,

Suppose there is a bump at the origin, is there a conformal mapping between the bumped half-space (y>|b-x|, |x|<b && y>0, |x|>b) and the flat upper half space (y>0)? Anyone has a hint? Thanks in advance.

Regards,
Tony

 

[mathwonk]

of course the answer is yes by riemann's theorem. to find an explicit one i suppose you could chop your region up into pieces and work on each piece. you have two quarter planes and a strip with a triangle removed, it looks like.

 

[radiofeda]

of course the answer is yes by riemann's theorem. to find an explicit one i suppose you could chop your region up into pieces and work on each piece. you have two quarter planes and a strip with a triangle removed, it looks like.

Thanks. I don't think I can chop the region up into pieces since that it is unbounded. I do know there is a conformal mapping cause we can imagine the coordinate grid lines in the bumped half space. However, I cannot find it although I know the boundary (line) shape (could be a piecewise function with respect to x or y).