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Unformatted text preview: he vector components are then the real and imaginary parts of this.
(B) Potential between two noncoaxial cylinders.
Find the potential between two cylinders C 1: z = 1 being grounded (potential 0) and
C2: z  0.4 = 0.4 having a potential of 110 volts.
This is hard to solve without some trick: First, consider the general FLT
z  z0
r(z) =
where c = z0– and z0 < 1.
–
cz  1
Then I claim that r maps the unit disc onto itself, but takes z0 to 0. The latter claim is
obvious. Let us check that first claim: Mapping the unit disc onto itself:
z = 1ﬁ z  z0 = z–  z0–
–
Since w = w– for every w
= z z–  z–0–
Since z = 1
= zz–  z z0–
–
= 1  z z–0–
Again, since z = 1
Therefore, r(z) = 1, as claimed.
Notice one further thing about this strange map: If we choose z0 to be real; z0 = b, say,
then
zb
r(z) =
bz  1
1b
2
and
r(1) =
= 1, and also r(1) =
= 1 so that r flips the unit circle over.
b1
2
Notice that, since this is an FLT, circles inside the unit disc must...
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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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