Complex Methods

Introduction
Analytic functions
	The complex plane and the Riemann sphere
	Complex differentiation
	Harmonic functions
	Multi-valued functions
	Mobius map
	Conformal maps
	Solving Laplace's equation using conformal maps
Contour integration and Cauchy's theorem
	Contour and integrals
	Cauchy's theorem
	Contour deformation
	Cauchy's integral formula
Laurent series and singularities
	Taylor and Laurent series
	Zeros
	Classification of singularities
	Residues
The calculus of residues
	The residue theorem
	Applications of the residue theorem
	Further applications of the residue theorem using rectangular contours
	Jordan's lemma
Transform theory
	Fourier transforms
	Laplace transform
	Elementary properties of the Laplace transform
	The inverse Laplace transform
	Solution of differential equations using the Laplace transform
	The convolution theorem for Laplace transforms