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Unformatted text preview: Complex Analysis Final Stephen Taylor June 23, 2005 1. For each of the following, determine whether the statement is true or false. Justify your answers with a brief explanation when your response is true and a counterexample when your response is false. (a) There is a branch cut for f ( z ) = log( z 2 + 1) so that f is analytic on { z C :  z  > 1 } . Solution: This statement is true. Since the branch points of a complex loga rithm function correspond to its singularities, we find the branch points to be i . We therefore take a branch cut along the segment [ i,i ], so f ( z ) is analytic everywhere exterior to the unit disc as desired. (b) If f has a pole of order m at a , then f /f has a pole of order 1 at a . Solution: This statement is true. If f has a pole or order m , then it has representation f ( z ) = a 1 ( a z ) m + a 2 ( a z ) m 1 + O ( z m +2 ) Differentiating we find f ( z ) = a 1 m ( a z ) m +1 + a 2 ( m 1) ( a z ) m + O ( z m +1 ) Using synthetic division, we find f ( z ) f ( z ) = m a z + a 2 a 1 + O ( z 1 ) from which we note that the quotient has a pole of order 1 at z = a . (c) If f ( z ) has an essential singularity at , then 1 /f (1 /z ) also has an essential singularity at . 1 Solution: This statement is true. Since f ( z ) has an essential singularity at , f ( z 1 ) has an essential singularity at zero. f ( z 1 ) = X a n z n We consider the nonterminating singular part of f ( z 1 ) given by f sing ( z 1 ) = a 1 z + a 2 z 2 + + a n z n + Using synthetic division, we find 1 f sing = z a 1 a 1 a 2 + O ( z 1 ) where O ( ) is the reverse order function indicating powers of z grow arbitrarily small. This tells us ( f ( z 1 )) 1 has an essential singularity. (d) If f is analytic and bounded on the unit disk, D , then f must be a rational function. Solution: This statement is false. Consider the entire function f ( z ) = e z . We find  e z  =  e x + iy  = e x which is bounded since x may only range between ( 1 , 1). Since f is clearly not a rational function, we conclude the invalidity of...
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This note was uploaded on 10/19/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.
 Three '09
 Cong
 Math

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