Introduction to complex analysis
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Contents

  • Introduction to differentiation
  • 1 Derivatives of complex functions
    • 1.1 Defining differentiable functions
    • 1.2 Combining differentiable functions
    • 1.3 Non-differentiability
    • 1.4 Higher-order derivatives
    • 1.5 A geometric interpretation of derivatives
    • 1.6 Further exercises
  • 2 The Cauchy–Riemann equations
    • 2.1 The Cauchy–Riemann theorems
    • 2.2 Proof of the Cauchy–Riemann Converse Theorem
    • 2.3 Further exercises
    • 2.4 Laplace’s equation and electrostatics
  • 3 Summary of Session 1
  • Introduction to integration
  • 1 Integrating real functions
    • 1.1 Areas under curves
    • 1.2 Integration on the real line
    • 1.3 Properties of the Riemann integral
    • 1.4 Introducing complex integration
  • 2 Integrating complex functions
    • 2.1 Integration along a smooth path
    • 2.2 Integration along a contour
    • 2.3 Reverse paths and contours
    • 2.4 Further exercises
  • 3 Evaluating contour integrals
    • 3.1 The Fundamental Theorem of Calculus
    • 3.2 Further exercises
  • 4 Summary of Session 2
  • Course conclusion
  • Acknowledgements

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