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An introduction to complex analysis and geometry / John P. D'Angelo.

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Author/Creator:
D'Angelo, John P.
Language:
English.
Publication date:
2010
Imprint:
Providence, R.I. : American Mathematical Society, 2010.
Format:
  • Book
  • xi, 163 p. ; 27 cm.
Bibliography:
Includes bibliographical references (p. 159-160) and index.
Contents:
  • Machine generated contents note: ch. 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
  • ch. 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
  • ch. 3 Complex Numbers and Geometry
  • 1. Lines, circles, and balls
  • 2. Analytic geometry
  • 3. Quadratic polynomials
  • 4. Linear fractional transformations
  • 5. The Riemann sphere
  • ch. 4 Power Series Expansions
  • 1. Geometric scries
  • 2. The radius of convergence
  • 3. Generating functions
  • 4. Fibonacci numbers
  • 5. An application of power series
  • 6. Rationality
  • ch. 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?
  • ch. 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
  • ch. 7 Applications of Complex Integration
  • 1. Singularities and residues
  • 2. Evaluating real integrals using complex variables methods
  • 3. Fourier transforms
  • 4. The Gamma function
  • ch. 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.
Series:
The Sally series
Pure and applied undergraduate texts ; 12.
Sally series (Providence, R.I.)
Subjects:
ISBN:
9780821852743
0821852744

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