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Complex Methods

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Author(s): R. E. Hunt, ed. Dexter Chua
Series: Cambridge Mathematical Tripos Part IB Lecture Notes
Publisher: University of Cambridge
Year: 2016

Language: English
City: Cambridge
Tags: maths; mathematics; math; advanced; college; university; higher; further; pure; applied; complex variable; complex analysis; complex numbers

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