Math 535 Problem Set 3
Page 108
1
1) Let z (t) = (1+ i)t with 0 t 1. Then xdz = 0 t(1+ i)dt = 1/2(1+ i).
6) Since |f (z ) 1| < 1 if we let w = f (z ) then log w is analytic since w is
never a negative real number. Thus we have log f (z ) is an analytic fu
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Homework 5, Math 535
page 133
1)
Consider the function f (z ) = z 2 + z . We have f (1/2) = 0 so the function is not
1 1 in a neighborhood of 1/2. Thus the largest radius circle about 0 in which it
could be 1 1 is 1/2.
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Now if f (z1 ) = f (z2 ) where z
Homework 6, Math 535
Page 148
4) On any simply connected domain not containing the origin f (z ) = 0
which means that log (z ) is analytic. But z = elog(z ) is analytic as well as
z z = ezlogz .
5) Any closed curve in the domain that winds around 1 also w
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