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Re: Conformal Mapping


Am 01.11.2012 08:22, schrieb MaxJ:
> Hi folks,
>
> I need help finding a Mobius transform such that the region:
>
> |z-i|<  sqrt(2)
> &&
> |z+i|<  sqrt(2)
>
> in z-plane be mapped conformally into a unit circle in w-plane.
>
> Any help is appreciated very much.

The boundaries are two circles with centers at +-i and radius^2 = 2. 
Consequently the contours are passing through the six points on the 
squares with vertices

+-1, 2i +-1  and +-1 -2i +-1

Consequently the lens shaped area in question is bounded by two 
symmetric quarter circles with center at +-i intersecting at +-1 with an 
angle of pi/2.

We conclude that the identity w=z is a conformal map of the lens into 
the inner of the unit circle.

The construction of a conformal map onto the unit circle, transforming 
to straight lines the pi/2 vertices at +-1, needs square roots centered 
at the vertices +-1.

This task could generate homework for up to one day approximately.

-- 

Roland Franzius



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