ALL 12 CHAPTERS COVERED
SOLUTIONS MANUAL
, Solutions Manual for
Functions of One Complex Variable I, Second Edition 1
© Copyright by Andreas Kleefeld, 2009
All Rights Reserved2
1 by John B. Conway
2 Last updated on January, 7th 2013
, PREFACE
Most of the exercises I solved were assigned homeworks in the graduate courses Math 713 and Math 714
”Complex Analysis I and II” at the University of Wisconsin – Milwaukee taught by Professor Dashan Fan
in Fall 2008 and Spring 2009.
The solutions manual is intented for all students taking a graduate level Complex Analysis course. Students
can check their answers to homework problems assigned from the excellent book “Functions of One Com-
plex Variable I”, Second Edition by John B. Conway. Furthermore students can prepare for quizzes, tests,
exams and final exams by solving additional exercises and check their results. Maybe students even study
for preliminary exams for their doctoral studies.
However, I have to warn you not to copy straight of this book and turn in your homework, because this
would violate the purpose of homeworks. Of course, that is up to you.
I strongly encourage you to send me solutions that are still missing to (LATEXpreferred
but not mandatory) in order to complete this solutions manual. Think about the contribution you will give
to other students.
If you find typing errors or mathematical mistakes pop an email to . The recent
version of this solutions manual can be found at
http://www.math.tu-cottbus.de/INSTITUT/kleefeld/Files/Solution.html.
The goal of this project is to give solutions to all of the 452 exercises.
CONTRIBUTION
I thank (without special order)
Christopher T. Alvin
Martin J. Michael
David Perkins
for contributions to this book.
2
, Contents
1 The Complex Number System 1
1.1 The real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The field of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 The complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Polar representation and roots of complex numbers . . . . . . . . . . . . . . . . . . . . . 5
1.5 Lines and half planes in the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 The extended plane and its spherical representation . . . . . . . . . . . . . . . . . . . . . 7
2 Metric Spaces and the Topology of C 9
2.1 Definitions and examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Sequences and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Elementary Properties and Examples of Analytic Functions 21
3.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Analytic functions as mappings. Möbius transformations . . . . . . . . . . . . . . . . . . 31
4 Complex Integration 42
4.1 Riemann-Stieltjes integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Power series representation of analytic functions . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Zeros of an analytic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 The index of a closed curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 Cauchy’s Theorem and Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 The homotopic version of Cauchy’s Theorem and simple connectivity . . . . . . . . . . . 63
4.7 Counting zeros; the Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . 66
4.8 Goursat’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Singularities 68
5.1 Classification of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 The Argument Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3
SOLUTIONS MANUAL
, Solutions Manual for
Functions of One Complex Variable I, Second Edition 1
© Copyright by Andreas Kleefeld, 2009
All Rights Reserved2
1 by John B. Conway
2 Last updated on January, 7th 2013
, PREFACE
Most of the exercises I solved were assigned homeworks in the graduate courses Math 713 and Math 714
”Complex Analysis I and II” at the University of Wisconsin – Milwaukee taught by Professor Dashan Fan
in Fall 2008 and Spring 2009.
The solutions manual is intented for all students taking a graduate level Complex Analysis course. Students
can check their answers to homework problems assigned from the excellent book “Functions of One Com-
plex Variable I”, Second Edition by John B. Conway. Furthermore students can prepare for quizzes, tests,
exams and final exams by solving additional exercises and check their results. Maybe students even study
for preliminary exams for their doctoral studies.
However, I have to warn you not to copy straight of this book and turn in your homework, because this
would violate the purpose of homeworks. Of course, that is up to you.
I strongly encourage you to send me solutions that are still missing to (LATEXpreferred
but not mandatory) in order to complete this solutions manual. Think about the contribution you will give
to other students.
If you find typing errors or mathematical mistakes pop an email to . The recent
version of this solutions manual can be found at
http://www.math.tu-cottbus.de/INSTITUT/kleefeld/Files/Solution.html.
The goal of this project is to give solutions to all of the 452 exercises.
CONTRIBUTION
I thank (without special order)
Christopher T. Alvin
Martin J. Michael
David Perkins
for contributions to this book.
2
, Contents
1 The Complex Number System 1
1.1 The real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The field of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 The complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Polar representation and roots of complex numbers . . . . . . . . . . . . . . . . . . . . . 5
1.5 Lines and half planes in the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 The extended plane and its spherical representation . . . . . . . . . . . . . . . . . . . . . 7
2 Metric Spaces and the Topology of C 9
2.1 Definitions and examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Sequences and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Elementary Properties and Examples of Analytic Functions 21
3.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Analytic functions as mappings. Möbius transformations . . . . . . . . . . . . . . . . . . 31
4 Complex Integration 42
4.1 Riemann-Stieltjes integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Power series representation of analytic functions . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Zeros of an analytic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 The index of a closed curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 Cauchy’s Theorem and Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 The homotopic version of Cauchy’s Theorem and simple connectivity . . . . . . . . . . . 63
4.7 Counting zeros; the Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . 66
4.8 Goursat’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Singularities 68
5.1 Classification of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 The Argument Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3
