Complex analysis
Dr John Arkinstall
3 points * First semester * 2 hours per week * Gippsland/Distance (odd-numbered years only) * Prerequisites: GAS1611 * Prohibitions: MAT3011
Objectives The objectives of this subject are for students to develop an understanding of the theory and techniques of calculus, applied to complex valued functions of a complex variable; understand the forms of representation of such functions, including multi-valued functions and series; become skilled in computational techniques involving complex functions and skilled in the choice of appropriate methods; become skilled in the computation of residues of complex functions at singularities, and in the interpretation of these in the context of various types of integrals and in finding inverse Laplace transforms.
Synopsis This subject develops the theory of functions of a complex variable and introduces diverse applications of this theory: complex sequences and series, functions of a complex variable, limits, continuity, points of discontinuity; differentiation of functions of a complex variable, singular points, the Cauchy-Riemann equation, harmonic functions; contours, line integrals, contour integration, Cauchy's theorem, Cauchy's integral formulas and related results; Power series, Taylor series, Laurent series, Taylor's theorem, Laurent's theorem, residues, the real integrals, inversion of Laplace transforms using the Bromwich integral formula; transformations, the bilinear transformation, conformal mapping, the Joukowski aerofoil; Laplace's equation in two independent variables, boundary value problems, Poisson's integral formulae for the circle and half-plane. For distance students, four two-hour problem-solving and expository classes are held over the semester, to supplement full notes, textbook, and assignments.
Assessment Assignments: 40% * Examination: 60%
Prescribed texts
Fisher S D Complex variables Wadsworth, 1986
Recommended texts
Ahlfors L V Complex analysis 3rd edn, McGraw-Hill, 1976
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