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Topic: conformal mapping (Read 8024 times) 

trusure
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conformal mapping
« on: Apr 19^{th}, 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
wu::riddles Moderator Uberpuzzler
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Re: conformal mapping
« Reply #1 on: Apr 19^{th}, 2009, 6:40pm » 
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The map z i(1z)/(1+z) takes the disk to the upperhalf plane, and the slit [a,1) to the slit (0, (1a)/(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 SchwarzChristoffel integral, with vertices at 0, i, and 0, and interior angles /2, 2, and /2, respectively, to map the upperhalf plane to this region. F(z) = i + _{0}^{z} w dw/[(w1)^{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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