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Topic: conformal mapping (Read 8201 times) |
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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
wu::riddles Moderator
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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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