This unfortunate name，which seems to imply that there is something unreal about these numbers and that they only lead a precarious existence in some people's imagination，has contributed much toward making the whole subject of complex numbers suspect in the eyes of generations of high school students.
　　-Zeev Nehari on the use of the term imaginary number
　　In the centuries prior to the movement of the 1800s to ensure that mathematical analysis was on solid logical footing，complex numbers，those numbers algebraically generated by adding √1 to the real field，were utilized with increasing frequency as an ever-growing number of mathematicians and physicists saw them as useful tools for solving problems of the time. The 19th century saw the birth of complex analysis，commonly referred to as function theory，as a field of study，and it has since grown into a beautiful and powerful subject.The functions referred to in the name "function theory" are primarily analytic functions，and a first course in complex analysis boils down to the study of the com-plex plane and the unique and often surprising properties of analytic functions. Fa-miliar concepts from calculus-the derivative，the integral，sequences and series-are ubiquitous in complex analysis，but their manifestations and interrelationships are novel in this setting.It is therefore possible，and arguably preferable，to see these topics addressed in a manner that helps stress these differences，rather than following the same ordering seen in calculus.

Preface
1 The Complex Numbers
1．1 Why?
1．2 The Algebra of Complex Numbers
1．3 The Geometry of the Complex Plane
1．4 The Topology of the Complex Plane
1．5 The Extended Complex Plane
1．6 Complex Sequences
1．7 Complex Series

2 Complex Functions and Mappings
2．1 Continuous Functions
2．2 Uniform Convergence
2．3 Power Series
2．4 Elementary Functions and Euler's Formula
2．5 Continuous Functions as Mappings
2．6 Linear Fractional Transformations
2．7 Derivatives
2．8 The Calculus of Real-Variable Functions
2．9 Contour Integrals

3 Analytic Functions
3．1 The Principle of Analyticity
3．2 Differentiable Functions are Analytic
3．3 Consequences of Goursat's Theorem
3．4 The Zeros of Analytic Functions
3．5 The Open Mapping Theorem and Maximum Principle
3．6 The Cauchy-Riemann Equations
3．7 Conformal Mapping and Local Univalence

4 Cauchy's Integral Theory
4．1 The Index of a Closed Contour
4．2 The Cauchy Integral Formula
4．3 Cauchy's Theorem

5 The Residue Theorem
5．1 Laurent Series
5．2 Classification of Singularities
5．3 Residues
5．4 Evaluation of Real Integrals
5．5 The Laplace Transform

6 Harmonic Functions and Fourier Series
6．1 Harmonic Functions
6．2 The Poisson Integral Formula
6．3 Further Connections to Analytic Functions
6．4 Fourier Series

Epilogue
A Sets and Functions
B Topics from Advanced Calculus
References
Index
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