Functions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.
Author(s): Roger Godement, P. Spain
Series: Universitext
Edition: 1
Publisher: Springer
Year: 2005
Language: English
Pages: 450
Analysis II......Page 2
Contents......Page 4
1 – Upper and lower integrals of a bounded function......Page 7
2 – Elementary properties of integrals......Page 11
3 – Riemann sums. The integral notation......Page 20
4 – Uniform limits of integrable functions......Page 22
5 – Application to Fourier series and to power series......Page 27
6 – The Borel-Lebesgue Theorem......Page 32
7 – Integrability of regulated or continuous functions......Page 35
8 – Uniform continuity and its consequences......Page 37
9 – Differentiation and integration under the sign......Page 42
10 – Semicontinuous functions15......Page 47
11 – Integration of semicontinuous functions......Page 54
12 – The fundamental theorem of the differential and integral calculus......Page 58
13 – Extension of the fundamental theorem to regulated functions......Page 65
14 – Convex functions; Holder and Minkowski inequalities......Page 71
15 – Integration by parts......Page 80
16 – The square wave Fourier series......Page 83
17 – Wallis’ formula......Page 86
18 – Taylor’s Formula......Page 88
19 – Change of variable in an integral......Page 97
20 – Integration of rational fractions......Page 101
21 – Convergent integrals: examples and definitions......Page 108
22 – Absolutely convergent integrals......Page 110
23 – Passage to the limit under the sign......Page 115
24 – Series and integrals......Page 121
25 – Differentiation under the sign......Page 124
26 – Integration under the sign......Page 130
27 – How to make C∞ a function which is not......Page 135
28 – Approximation by polynomials......Page 141
29 – Functions having given derivatives at a point......Page 144
30 – Radon measures on a compact set......Page 147
31 – Measures on a locally compact set......Page 156
32 – The Stieltjes construction......Page 163
33 – Application to double integrals......Page 170
34 – Definition and examples......Page 174
35 – Derivatives of a distribution......Page 179
Appendix to Chapter V Introduction to the Lebesgue Theory......Page 184
1 – Comparison relations......Page 200
2 – Rules of calculation......Page 202
3 – Truncated expansions......Page 203
4 – Truncated expansion of a quotient......Page 205
5 – Gauss’ convergence criterion......Page 207
6 – The hypergeometric series......Page 209
7 – Asymptotic study of the equation xex = t......Page 211
8 – Asymptotics of the roots of sin x. log x = 1......Page 213
9 – Kepler’s equation......Page 215
10 – Asymptotics of the Bessel functions......Page 218
11 – Cavalieri and the sums 1k + 2k + . . . + nk......Page 229
12 – Jakob Bernoulli......Page 231
13 – The power series for cot z......Page 236
14 – Euler and the power series for arctan x......Page 239
15 – Euler, Maclaurin and their summation formula......Page 243
16 – The Euler-Maclaurin formula with remainder......Page 244
17 – Calculating an integral by the trapezoidal rule......Page 246
18 – The sum 1+1/2 + . . . + 1/n, the infinite product for the Γ function, and Stirling’s formula......Page 247
19 – Analytic continuation of the zeta function......Page 252
1 – Cauchy’s integral formula for a circle......Page 255
2 – Functions and measures on the unit circle......Page 259
3 – Fourier coefficients......Page 265
4 – Convolution product on T......Page 270
5 – Dirac sequences in T......Page 274
6 – Absolutely convergent Fourier series......Page 278
7 – Hilbertian calculations......Page 279
8 – The Parseval-Bessel equality......Page 281
9 – Fourier series of differentiable functions......Page 287
10 – Distributions on T......Page 291
11 – Dirichlet’s theorem......Page 299
12 – Fejer’s theorem......Page 305
13 – Uniformly convergent Fourier series......Page 307
§ 4. Analytic and holomorphic functions......Page 311
14 – Analyticity of the holomorphic functions......Page 312
15 – The maximum principle......Page 314
16 – Functions analytic in an annulus. Singular points. Meromorphic functions......Page 317
17 – Periodic holomorphic functions......Page 323
18 – The theorems of Liouville and of d’Alembert-Gauss......Page 324
19 – Limits of holomorphic functions......Page 334
20 – Infinite products of holomorphic functions......Page 336
21 – Analytic functions defined by a Cauchy integral......Page 344
22 – Poisson’s function......Page 346
23 – Applications to Fourier series......Page 348
24 – Harmonic functions......Page 351
25 – Limits of harmonic functions......Page 355
26 – The Dirichlet problem for a disc......Page 358
27 – The Poisson summation formula......Page 361
28 – Jacobi’s theta function......Page 365
29 – Fundamental formulae for the Fourier transform......Page 369
30 – Extensions of the inversion formula......Page 373
31 – The Fourier transform and differentiation......Page 378
32 – Tempered distributions......Page 382
§ 1. How to fool young innocents......Page 391
§ 2. The evolution of R&D funding in America......Page 412
§ 3. Applied mathematics in America......Page 428
Index......Page 440
Table of Contents of Volume I......Page 444
Universitext......Page 447
