Complex Analysis

URL study guide

Description

• • • Complex numbers: Algebraic properties, fundamental topological concepts, sequences, continuity, series and analytic functions.
    • Holomorphic Functions: Complex differentiability, Cauchy-Riemann, harmonic functions
    • Line integrals: Curves, integrals along curves, connected sets.
    • Cauchy Integral Theorem, Theorem of Goursat, homotopy of curves and invariance of complex integrals
    • Implications of the Cauchy Integral Theorem: Cauchy integral formula, Mean value Theorem, Taylor’s theorem, Liouville’s theorem, Fundamental theorem of Algebra, Identity theorem.
    • Laurent series and analysis of isolated singularities.
    • Residue calculus: Residue formula and its application to real and complex integrals and to Fourier and Laplace transforms.
    • Some basic applications of complex analysis, for example in electromagnetism, fluid dynamics, or image processing.

Objectives

• recall and state the definitions and properties in the theory of complex (holomorphic/analytic) functions and power series, in particular related to the exponential function, the logarithm, and the trigonometric functions;
• compute and understand the meaning of derivatives in the setting of complex functions; relate complex differentiable functions to the Cauchy- Riemann equations for real functions of two real variables;
• determine (regions of) convergence/uniform convergence of power series and related series, derivatives of such series, and determine closed expressions in fairly straightforward situations;
• determine the existence of antiderivatives for holomorphic functions and their domain;
• determine Taylor and Laurent series of functions around given points;
• determine the character of singular points;
• deduce (global) properties of holomorphic functions based on their (Tay- lor/Laurent) series representation around given points;
• compute residues and understand their role in computing (real/complex) integrals;
• solve real and complex integration problems, including Fourier and Laplace transforms.

Method of Assessment