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Contents

• V Full version
• 0 Introduction
• 1 Analytic functions
• 1.1 The complex plane and the Riemann sphere
• 1.2 Complex differentiation
• 1.3 Harmonic functions
• 1.4 Multi-valued functions
• 1.5 Mobius map
• 1.6 Conformal maps
• 1.7 Solving Laplace's equation using conformal maps
• 2 Contour integration and Cauchy's theorem
• 2.1 Contour and integrals
• 2.2 Cauchy's theorem
• 2.3 Contour deformation
• 2.4 Cauchy's integral formula
• 3 Laurent series and singularities
• 3.1 Taylor and Laurent series
• 3.2 Zeros
• 3.3 Classification of singularities
• 3.4 Residues
• 4 The calculus of residues
• 4.1 The residue theorem
• 4.2 Applications of the residue theorem
• 4.3 Further applications of the residue theorem using rectangular contours
• 4.4 Jordan's lemma
• 5 Transform theory
• 5.1 Fourier transforms
• 5.2 Laplace transform
• 5.3 Elementary properties of the Laplace transform
• 5.4 The inverse Laplace transform
• 5.5 Solution of differential equations using the Laplace transform
• 5.6 The convolution theorem for Laplace transforms