Then use z to go to a simpler region follow by

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Unformatted text preview: map to circles inside the unit disc (they can hardly map to infinite lines!) and it certainly looks like circles 34 centered at z0 inside the disc map to circles centered at 0 (look at very small circles, for instance). Now back to the example at hand: We try to adjust this so that the non-concentric circles are moved onto the concentric circles |z| = 1 and |z| = r for some r < 1. For this, we take z0 = b, a point somewhere on the x-axis in order to map the off-centered inner circle onto the circle centered at 0 radius r. Since b = b, we have — z-b r(z) = bz - 1 We would also like 0 to map to r and 0.8 to map to - r (remember the flipping effect—draw a picture). -b r= giving b = r -1 0.8 - b -r = 0.8b - 1 Substituting the first in the second gives, after some fiddling, the quadratic 2 2b - 5b + 2 = 0 (b - 2)(2b - 1) = 0 b = 2 (no good; this will give r = 2 -- too big) and b = 0.5, which we use. Therefore, our FLT is z - 0.5 2z - 1 w = r(z) = = 0.5z - 1 z-2 This happens to take the inner circle into a cir...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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