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Complex Analysis: A Modern First Course in Function Theory

Jerry R. Muir, Jr.
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is a brief text for a one-semester first course in complex analysis, aimed at students who have a good understanding of calculus and some understanding of real analysis. The approach is through power series, with contour integrals and the Cauchy–Riemann equations downplayed.

The book makes a serious effort to give a reasonable introduction to complex analysis within one semester, and does this primarily by being very selective about topics and by not going into much depth. This selectivity sometimes has the bad effect of introducing topics that don’t go anywhere. For example, the stereographic projection (one-point compactification) of the complex plane is defined on p. 16. It is awkwardly placed, as it is after the discussion of the topology of the complex plane, which was done without taking this compactification into account. It seems to be there only to justify having a single infinity for the complex numbers, rather than the plus and minus infinity we are used to in the reals. This extended complex plane is then used without comment in the rest of the book. Another example is that there are several pages on linear fractional transformations, and several pages on conformal mapping, but the connection between the two is only alluded to briefly in an example and not developed. The book doesn’t cover very many types of functions: just polynomials, exponential, sine and cosine; and rational combinations of these, so interesting special functions such as the gamma function and the Riemann zeta function are not mentioned.

The book has a large number of worked examples, that show a lot of variety. The exercises are numerous and reasonable but not very difficult; most of them are almost drill and ask the student to work problems that are very similar to the worked examples (and in many cases even tell the student which example to use as a model). Some of the exercises are more challenging and introduce topics not covered in the main narrative. No answers are given in the book for the exercises, which would be a drawback for many students.

Very Good Feature: extensive end-of-section notes with history and background. Most of this will go over the heads of the students, but instructors and people using it as a reference will appreciate it.

The book’s marketing is inaccurate. The book is subtitled “A Modern First Course in Function Theory”, but mathematically and pedagogically there’s nothing here less than a century old. The newest mathematical things are some point-set topology and a few mentions of automorphisms and of univalent (schlicht) functions. The book’s front and back covers trumpet “with website”, but this is not mentioned inside the book, and the only web resource on the publisher site is a downloadable instructor solutions manual.

Bottom line: a competent treatment, but nothing special. A similar but more comprehensive and better-executed book is Bak & Newman’s Complex Analysis.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

Preface ix

1 The Complex Numbers 1

1.1 Why? 1

1.2 The Algebra of Complex Numbers 3

1.3 The Geometry of the Complex Plane 7

1.4 The Topology of the Complex Plane 9

1.5 The Extended Complex Plane 16

1.6 Complex Sequences 18

1.7 Complex Series 24

2 Complex Functions and Mappings 29

2.1 Continuous Functions 29

2.2 Uniform Convergence 34

2.3 Power Series 38

2.4 Elementary Functions and Euler’s Formula 43

2.5 Continuous Functions as Mappings 50

2.6 Linear Fractional Transformations 53

2.7 Derivatives 64

2.8 The Calculus of Real Variable Functions 70

2.9 Contour Integrals 75

3 Analytic Functions 87

3.1 The Principle of Analyticity 87

3.2 Differentiable Functions are Analytic 89

3.3 Consequences of Goursat’s Theorem 100

3.4 The Zeros of Analytic Functions 104

3.5 The Open Mapping Theorem and Maximum Principle 108

3.6 The Cauchy–Riemann Equations 113

3.7 Conformal Mapping and Local Univalence 117

4 Cauchy’s Integral Theory 127

4.1 The Index of a Closed Contour 127

4.2 The Cauchy Integral Formula 133

4.3 Cauchy’s Theorem 139

5 The Residue Theorem 145

5.1 Laurent Series 145

5.2 Classification of Singularities 152

5.3 Residues 158

5.4 Evaluation of Real Integrals 165

5.5 The Laplace Transform 174

6 Harmonic Functions and Fourier Series 183

6.1 Harmonic Functions 183

6.2 The Poisson Integral Formula 191

6.3 Further Connections to Analytic Functions 201

6.4 Fourier Series 210

Epilogue 227

A Sets and Functions 239

B Topics from Advanced Calculus 247

References 255

Index 257