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        general >> complex analysis >> conformal mapping
        
(Message started by: trusure on Apr 19^th, 2009, 9:52am)

Title: conformal mapping
Post by trusure on Apr 19^th, 2009, 9:52am

I'm trying to construct a conformal map from
D(0,1)\[a,1) to D(0,1) , where 0<a<1 , but it didn't work with me!

if it is from D(0,1) to D(0,1) it is easy, but here ??!

can anyone help me, actually,I have a problem with understanding how I can construct such mapping !?

Thank you

Title: Re: conformal mapping
Post by Eigenray on Apr 19^th, 2009, 6:40pm

The map z http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mapsto.gif i(1-z)/(1+z) takes the disk to the upper-half plane, and the slit [a,1) to the slit (0, (1-a)/(1+a)i ]. For simplicity scale so that the slit becomes (0, i].

Now we basically have a polygonal region so we can use a Schwarz-Christoffel integral, with vertices at 0, i, and 0, and interior angles http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/2, 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif, and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/2, respectively, to map the upper-half plane to this region.

F(z) = i + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif₀^z w dw/[(w-1)^1/2 (w+1)^1/2]

does the trick (taking the branch of sqrt which is positive on the positive real axis).