Course Outline
Complex Methods
COMPLEX METHODS
Analytic functions
- Definition of an analytic function. Cauchy-Riemann equations. Analytic functions as conformal mappings; examples. Application to the solutions of Laplace’s equation in various domains. Discussion of logz and za. [6]
Contour integration and Cauchy’s Theorem[Proofs of theorems in this section will not be examined in this course.]
- Contours, contour integrals. Cauchy’s theorem and Cauchy’s integral formula. Taylor and Laurent series. Zeros, poles and essential singularities. [4]
Residue calculus
- Residue theorem, calculus of residues. Jordan’s lemma. Evaluation of definite integrals by contour integration. [3]
Fourier and Laplace transforms
- Laplace transform: definition and basic properties; inversion theorem (proof not required); convolution theorem. Examples of inversion of Fourier and Laplace transforms by contour integration. Applications to differential equations. [3]
Appropriate books
- M.J. Ablowitz and A.S. Fokas Complex variables: introduction and applications. CUP 2003
- G.B. Arfken,H.J. Weber & F.E. Harris Mathematical Methods for Physicists. Elsevier 2013
- G.J.O. Jameson A First Course in Complex Functions. Chapman and Hall 1970
- T. Needham Visual complex analysis. Clarendon 1998
- † H.A. Priestley Introduction to Complex Analysis. Clarendon 1990
- † K. F. Riley, M. P. Hobson, and S.J. Bence Mathematical Methods for Physics and Engineering: a comprehensive guide. Cambridge University Press 2002
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