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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
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
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).
giving b = r
0.8 - b
0.8b - 1
Substituting the first in the second gives, after some fiddling, the quadratic
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
This happens to take the inner circle into a cir...
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