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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 nonconcentric 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 xaxis in order to map the offcentered inner circle onto
the circle centered at 0 radius r. Since b = b, we have
—
zb
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
z2
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.
 Fall '03
 StefanWaner
 Math, Algebra, Geometry, Complex Numbers

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