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# final - Complex Analysis Final Stephen Taylor 1 For each of...

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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

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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 ) = -∞ a n z n We consider the non-terminating 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 the statement.
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final - Complex Analysis Final Stephen Taylor 1 For each of...

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