Introduction to complex analysis

This free course, Introduction to complex analysis, is an extract from the third level Open University course M337 Complex analysis, which develops the theory of functions of a complex variable, emphasising their geometric properties and indicating some applications.Introduction to complex analysis consists of two sessions. The first session covers differentiation, and the second session covers integration of complex functions. The course should take about 12 hours to study. This content forms part of the Dangoor Education collection, the educational arm of The Exilarch's Foundation.

• Session1Session 1: Differentiation
• 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
• Acknowledgements
• Session2Session 2: Integration
• 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