Department
of Mathematics, U. of Rhode Island

Instructor |
Orlando
Merino, merino@math.uri.edu, 874-4442, Tyler Hall 220 |

Term |
Spring
2006 |

Text |
Functions of One Complex Variable, by John B. Conway (Second
Ed.) ISBN: 0-387-90328-3 |

Topics |
We shall cover chapters I--V of the text, and sections 1,2
of VI if time permits: History, The complex number system (the complex plane, roots of complex numbers, the extended plane and stereographic projection), Metric Spaces and the topology of C (compactness, continuity and uniform convergence) Elementary properties of analytic functions (power series, analytic functions, Cauchy Riemann Equations, branches of the logarithm, linear fractional transformations, conformal functions), complex integration (Riemann Stieltjes integrals, zeros of analytic functions, Liouville's Theorem, The Fundamental Theorem of Algebra, the index of a closed curve, Cauchy's theorem, the open mapping theorem, Goursat's theorem), Singularities (Laurent expansions, classification of singularities, Casorati-Weierstrass Theorem, residues, the argument principle, Rouche's Theorem). The Maximum Modulus Theorem and Schwarz's lemma. |

Evaluation |
Evaluation
is based on weekly homework (50%), a midterm exam (25 %), and a final exam
(25 %). |

On the course |
We shall develop the theory
of complex functions in a mathematically rigorous way. |

Additional Information |
A Short
History of Complex Numbers (pdf document, 5 pages) timeline from Wolfram Research |