Gowers W.T. Further Mathematical Analysis (53p)

Gowers W.T. Further Mathematical Analysis (53p) - Further...

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Unformatted text preview: Further Analysis Prof. W.T. Gowers Lent 1997 These notes are maintained by Paul Metcalfe. Comments and corrections to pdm23@cam.ac.uk . Revision: 2.9 Date: 2004/07/26 07:28:53 The following people have maintained these notes. – date Paul Metcalfe Contents Introduction v 1 Topological Spaces 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Building New Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Compactness 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Some compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Consequences of compactness . . . . . . . . . . . . . . . . . . . . . 7 2.4 Other forms of compactness . . . . . . . . . . . . . . . . . . . . . . 7 3 Connectedness 9 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Connectedness in R . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Path connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Preliminaries to complex analysis 13 4.1 Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Complex Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.4 Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Cauchy’s theorem and its consequences 19 5.1 Cauchy’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.3 Consequences of Cauchy’s Theorem . . . . . . . . . . . . . . . . . . 23 6 Power Series 27 6.1 Analyticity and Holomorphy . . . . . . . . . . . . . . . . . . . . . . 27 6.2 Classification of Isolated Singularities . . . . . . . . . . . . . . . . . 31 7 Winding Numbers 35 7.1 Introduction and Definition . . . . . . . . . . . . . . . . . . . . . . . 35 7.2 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8 Cauchy’s Theorem (homology version) 41 iii iv CONTENTS Introduction These notes are based on the course “Further Analysis” given by Prof. W.T. Gowers 1 in Cambridge in the Lent Term 1997. These typeset notes are totally unconnected with Prof. Gowers. Other sets of notes are available for different courses. At the time of typing these courses were: Probability Discrete Mathematics Analysis Further Analysis Methods Quantum Mechanics Fluid Dynamics 1 Quadratic Mathematics Geometry Dynamics of D.E.’s Foundations of QM Electrodynamics Methods of Math. Phys Fluid Dynamics 2 Waves (etc.) Statistical Physics General Relativity Dynamical Systems Combinatorics Bifurcations in Nonlinear Convection They may be downloaded from http://www.istari.ucam.org/maths/ ....
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Gowers W.T. Further Mathematical Analysis (53p) - Further...

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