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See discussions, stats, and author profiles for this publication at: Complex Analysis: Problems with solutions Book · August 2016 CITATIONS 0 READS 304,567 1 author: Some of the authors of this publication are also working on these related projects: ABC of Mathematics: An interactive experience View project Complex Analysis: A visual and interactive introduction View project Juan Carlos Ponce Campuzano The University of Queensland 46 PUBLICATIONS 42 CITATIONS SEE PROFILE All content following this page was uploaded by Juan Carlos Ponce Campuzano on 01 June 2019. The user has requested enhancement of the downloaded file.
Complex Analysis Problems with solutions Juan Carlos Ponce Campuzano
Copyright c 2016 Juan Carlos Ponce Campuzano P UBLISHED BY J UAN C ARLOS P ONCE C AMPUZANO ISBN 978-0-6485736-1-6 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. License at: First e-book version, August 2015
Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 Basic algebraic and geometric properties 7 1.2 Modulus 10 1.3 Exponential and Polar Form, Complex roots 13 2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1 Basic notions 19 2.2 Limits, Continuity and Differentiation 27 2.3 Analytic functions 31 2.3.1 Harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3 Complex Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 Contour integrals 39 3.2 Cauchy Integral Theorem and Cauchy Integral Formula 43 3.3 Improper integrals 56 4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1 Taylor and Laurent series 59 4.2 Classification of singularities 68
4.3 Applications of residues 74 4.3.1 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Foreword This text constitutes a collection of problems for using as an additional learning resource for those who are taking an introductory course in complex analysis. The problems are numbered and allocated in four chapters corresponding to different subject areas: Complex Numbers , Functions , Complex Integrals and Series . The majority of problems are provided with answers, detailed procedures and hints (sometimes incomplete solutions). Of course, no project such as this can be free from errors and incompleteness. I will be grateful to everyone who points out any typos, incorrect solutions, or sends any other suggestion for improving this manuscript. Contact: [email protected] 2016
1. Complex Numbers 1.1 Basic algebraic and geometric properties 1. Verify that (a) 2 - i - i 1 - 2 i = - 2 i (b) ( 2 - 3 i )( - 2 + i ) = - 1 + 8 i Solution. We have 2 - i - i 1 - 2 i = 2 - i - i + 2 = - 2 i , and ( 2 - 3 i )( - 2 + i ) = - 4 + 2 i + 6 i - 3 i 2 = - 4 + 3 + 8 i = - 1 + 8 i . 2. Reduce the quantity 5 i ( 1 - i )( 2 - i )( 3 - i ) to a real number. Solution. We have 5 i ( 1 - i )( 2 - i )( 3 - i ) = 5 i ( 1 - i )( 5 - 5 i ) = i ( 1 - i ) 2 = i - 2 i = 1 2
8 Chapter 1. Complex Numbers 3. Show that (a) Re ( iz ) = - Im ( z ) ; (b) Im ( iz ) = Re ( z ) . Proof. Let z = x + yi with x = Re ( z ) and y = Im ( z ) . Then Re ( iz ) = Re ( - y + xi ) = - y = - Im ( z ) and Im ( iz ) = Im ( - y + xi ) = x = Re ( z ) . 4. Verify the associative law for multiplication of complex numbers. That is, show that ( z 1 z 2 ) z 3 = z 1 ( z 2 z 3 ) for all z 1 , z 2 , z 3 C .

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