Calculus of Residua - Complex Functions Theory a-2

Cover......Page 1
Title......Page 2
ISBN 978-87-7681-691-9......Page 3
Contents......Page 4
Introduction......Page 7
1.1 Accumulations points and limes superio......Page 9
1.2 Power series......Page 11
1.3 Expansion of an analytic function in a power series......Page 22
1.4 Linear differential equations......Page 29
1.5 Zeros of analytical functions......Page 40
1.6 Simple Fourier series......Page 44
1.7 The maximum principle......Page 48
2.1 Harmonic functions......Page 55
2.2 The maximum principle for harmonic functions......Page 59
2.3 The biharmonic equation......Page 61
2.4 Poisson’s Integral Formula......Page 64
2.5 Electrostatic fields......Page 67
2.6 Static temperature fields......Page 69
3.1 Laurent series......Page 71
3.2 Fourier series II......Page 79
3.3 Solution of a linear differential equation by means of Laurent series......Page 80
3.4 Isolated boundary points......Page 84
3.5 Infinity  as an isolated boundary point......Page 89
3.6 Residua......Page 93
3.7 Simple rules of computation of the residuum at a (finite) pole......Page 97
3.8 The residuum at infinity......Page 102
3.9 Summary of the Calculus of Residua......Page 108
4.1 Trigonometric integrals......Page 111
4.2 Improper integrals......Page 113
4.3 Cauchy’s principal value......Page 124
4.4 The Mellin transform......Page 128
4.5 Residuum formulæ for sums of series......Page 133
Index......Page 139