Preface Chapter 1．From the Real Numbers to the Complex Numbers 1．Introduction 2．Number systems 3．Inequalities and ordered fields 4．The complex numbers 5．Alternative definitions of C 6．A glimpse at metric spaces Chapter 2．Complex Numbers 1．Complex conjugation 2．Existence of square roots 3．Limits 4．Convergent infinite series 5．Uniform convergence and consequences 6．The unit circle and trigonometry 7．The geometry of addition and multiplication 8．Logarithms Chapter 3．Complex Numbers and Geometry 1．Lines， circles， and balls 2．Analytic geometry 3．Quadratic polynomials 4．Linear fractional transformations 5．The Riemann sphere Chapter 4．Power Series Expansions 1．Geometric series 2．The radius of convergence 3．Generating functions 4．Fibonacci numbers 5．An application of power series 6．Rationality Chapter 5．Complex Differentiation 1．Definitions of complex analytic function 2．Complex differentiation 3．The Cauchy-Riemann equations 4．Orthogonal trajectories and harmonic functions 5．A glimpse at harmonic functions 6．What is a differential form? Chapter 6．Complex Integration 1．Complex-valued functions 2．Line integrals 3．Goursat's proof 4．The Cauchy integral formula 5．A return to the definition of complex analytic function Chapter 7．Applications of Complex Integration 1．Singularities and residues 2．Evaluating real integrals using complex variables methods 3．Fourier transforms 4．The Gamma function Chapter 8．Additional Topics 1．The minimum-maximum theorem 2．The fundamental theorem of algebra 3．Winding numbers， zeroes， and poles 4．Pythagorean triples 5．Elementary mappings 6．Quaternions 7．Higher-dimensional complex analysis Fhrther reading Bibliography Index
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