7 ln z 2 ln05 the real potential is 2z 1 1587

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Unformatted text preview: he vector components are then the real and imaginary parts of this. (B) Potential between two non-coaxial 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 z-b r(z) = bz - 1 1-b -2 and r(1) = = -1, and also r(-1) = = 1 so that r flips the unit circle over. b-1 -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.

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