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Unformatted text preview: e ¯ z is not analytic. (b) Let f ( z ) be an analytic function of z . Show that the function ¯ f deﬁned by ¯ f ( z ) = f (¯ z ) is also an analytic function of z . 4. (20 pts.) Show that the functions deﬁned by f ( z ) = log | z | + i arg( z ) and f ( z ) = p | z | e i arg( z ) / 2 are holomorphic (analytic) in the sector | z | > 0, | arg( z ) | < π . What are the corresponding derivatives df/dz ? 5. (30 pts.) Show that the following functions are harmonic. For each one of them ﬁnd its harmonic conjugate and form the corresponding holomorphic function. (a) u ( x,y ) = x ln( r )-y arctan( y x ) ( r 6 = 0). (b) u ( x,y ) = arg( z ) ( | arg( z ) | < π , r 6 = 0). (c) u ( x,y ) = r n cos( nθ ). (d) u ( x,y ) = y/r 2 ( r 6 = 0). Due at 5pm on Wed. October 21....
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This note was uploaded on 01/14/2010 for the course ACM 1 taught by Professor Prof during the Fall '09 term at Caltech.
- Fall '09