Mathematics
MA287: Analysis II (5 ECTS)
This module introduces the theory of functions of a complex variable, statring with an introduction to complex numbers and ending with applications of the Residue Theorem and conformal transformations.
Taught in Semester(s) 2. Examined in Semester(s) 2.
Workload: 100 hours (24 Lecture hours, 12 Tutorial hours, 64 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- represent a complex number as a point in the plane; calculate the modulus and argument of a complex number; switch between Cartesian and polar forms; calculate the $n$th roots of a complex number;
- decide where a function is differentiable (resp. analytic) using the Cauchy-Riemann equations;
- calculate the complex derivative of a function; decide whether a function is harmonic; calculate the harmonic conjugate of a harmonic function;
- do various calculations involving exponentials and logarithms;
- parameterize a variety of paths in the plane;
- calculate the integral of a function along a given path;
- apply Cauchy's Theorem to compute integrals; apply Cauchy's Integral Formula to calculate various integrals;
- calculate the Taylor series of a variety of elementary functions;
- deduce the Laurent series of a range of functions;
- apply the Residues Theorem to calculate various improper integrals.
Indicative Content
- Part 1. Complex numbers. Complex numbers and their representation. Modulus and argument. Polar form. Roots.
- Part 2. Derivatives. Complex functions. Cauchy-Riemann equations. Harmonic functions and harmonic conjugates. Elementary functions.
- Part 3. Integrals. Line integrals: the Fundamental Theorem of Calculus. Cauchy's Theorem and its consequences. Cauchy's Integral Formula. Applications: Liouville's Theorem and the Fundamental Theorem of Algebra.
- Part 4. Series. Applications to real analysis.Taylor series. Laurent series. The Residues Theorem. Applications to improper integrals.
Module Resources
- Complex Variables by S. Lipschutz et al. Schaum's
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