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   Author  Topic: conformal mapping  (Read 7919 times)
trusure
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conformal mapping  
« on: Apr 19th, 2009, 9:52am »
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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
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Eigenray
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Re: conformal mapping  
« Reply #1 on: Apr 19th, 2009, 6:40pm »
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The map z 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 /2, 2,  and /2, respectively, to map the upper-half plane to this region.
 
F(z) = i + 0z 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).
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