final - Complex Analysis Final Stephen Taylor June 23, 2005...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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...
View Full Document

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.

Page1 / 7

final - Complex Analysis Final Stephen Taylor June 23, 2005...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online