A Course in Mathematical Analysis

✍ Scribed by Garling, D J H

• Publisher: Cambridge University Press Textbooks
• Year: 2014;2013
• Tongue: English
• Leaves: 332
• Category: Library

✦ Table of Contents

Cover......Page 0
Half-title......Page 4
Title page......Page 6
Copyright information......Page 7
Table of contents......Page 8
Introduction......Page 12
Part Five Complex analysis......Page 14
20.1 Holomorphic functions......Page 16
20.2 The Cauchy–Riemann equations......Page 19
20.3 Analytic functions......Page 24
20.4 The exponential, logarithmic and circular functions......Page 30
20.5 Infinite products......Page 34
20.6 The maximum modulus principle......Page 35
21.1 Winding numbers......Page 39
21.2 Homotopic closed paths......Page 44
21.3 The Jordan curve theorem......Page 50
21.4 Surrounding a compact connected set......Page 56
21.5 Simply connected sets......Page 59
22.1 Integration along a path......Page 63
22.2 Approximating path integrals......Page 69
22.3 Cauchy's theorem......Page 73
22.4 The Cauchy kernel......Page 78
22.5 The winding number as an integral......Page 79
22.6 Cauchy's integral formula for circular and square paths......Page 81
22.7 Simply connected domains......Page 87
22.8 Liouville's theorem......Page 88
22.9 Cauchy's theorem revisited......Page 89
22.10 Cycles; Cauchy's integral formula revisited......Page 91
22.11 Functions defined inside a contour......Page 93
22.12 The Schwarz reflection principle......Page 94
23.1 Zeros......Page 97
23.2 Laurent series......Page 99
23.3 Isolated singularities......Page 102
23.4 Meromorphic functions and the complex sphere......Page 107
23.5 The residue theorem......Page 109
23.6 The principle of the argument......Page 113
23.7 Locating zeros......Page 119
24.1 Calculating residues......Page 122
24.2 Integrals of the form ∫[sup(2π)][sub(0)] f(cos t, sin t) dt......Page 123
24.3 Integrals of the form ∫[sup(∞)][sub(-∞)] f(x) dx......Page 125
24.4 Integrals of the form ∫[sup(∞)][sub(0)] x[sup(α)] f(x) dx......Page 131
24.5 Integrals of the form ∫[sup(∞)][sub(0)] f(x) dx......Page 134
25.1 Introduction......Page 138
25.3 Univalent functions on the punctured plane C......Page 139
25.4 The Möbius group......Page 140
25.5 The conformal automorphisms of D......Page 147
25.6 Some more conformal transformations......Page 148
25.7 The space H(U) of holomorphic functions on a domain U......Page 152
25.8 The Riemann mapping theorem......Page 154
26.1 Jensen's formula......Page 157
26.2 The function π cot πz......Page 159
26.3 The functions π cosec πz......Page 161
26.4 Infinite products......Page 164
26.5 Euler's product formula*......Page 167
26.6 Weierstrass products......Page 172
26.7 The gamma function revisited......Page 179
26.8 Bernoulli numbers, and the evaluation of ζ(2k)......Page 183
26.9 The Riemann zeta function revisited......Page 186
Part Six Measure and Integration......Page 190
27.1 Introduction......Page 192
27.2 The size of open sets, and of closed sets......Page 193
27.3 Inner and outer measure......Page 197
27.4 Lebesgue measurable sets......Page 199
27.5 Lebesgue measure on R......Page 201
27.6 A non-measurable set......Page 203
28.1 Some collections of sets......Page 206
28.2 Borel sets......Page 209
28.3 Measurable real-valued functions......Page 211
28.4 Measure spaces......Page 214
28.5 Null sets and Borel sets......Page 218
28.6 Almost sure convergence......Page 219
29.1 Integrating non-negative functions......Page 223
29.2 Integrable functions......Page 228
29.3 Changing measures and changing variables......Page 235
29.4 Convergence in measure......Page 237
29.5 The spaces L[sup(1)][sub(R)] (X, Σ, μ) and L[sup(1)][sub(C)] (X, Σ, μ)......Page 243
29.6 The spaces L[sup(p)][sub(R)] (X, Σ, μ) and L[sup(p)][sub(C)] (X, Σ, μ), for 0 < p < ∞......Page 245
29.7 The spaces L[sup(∞ )][sub(R)] (X, Σ, μ) and L[sup(∞)][sub(C)] (X, Σ, μ)......Page 252
30.1 Outer measures......Page 254
30.2 Caratheodory's extension theorem......Page 257
30.3 Uniqueness......Page 260
30.4 Product measures......Page 262
30.5 Borel measures on R, I......Page 269
31.1 Signed measures......Page 273
31.2 Complex measures......Page 278
31.3 Functions of bounded variation......Page 280
32.1 Borel measures on metric spaces......Page 285
32.2 Tight measures......Page 287
32.3 Radon measures......Page 289
33.1 The Lebesgue decomposition theorem......Page 292
33.2 Sublinear mappings......Page 295
33.3 The Lebesgue differentiation theorem......Page 297
33.4 Borel measures on R, II......Page 301
34.1 Bernstein polynomials......Page 304
34.2 The dual space of L[sup(p)][sub(C)] (X, Σ, μ), for 1 ≤ p < ∞......Page 307
34.3 Convolution......Page 308
34.4 Fourier series revisited......Page 313
34.5 The Poisson kernel......Page 316
34.6 Boundary behaviour of harmonic functions......Page 323
Index......Page 325
Contents for Volume I......Page 329
Contents for Volume II......Page 332