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: 451
Analysis II......Page 3
Contents......Page 5
1 – Upper and lower integrals of a bounded function......Page 8
2 – Elementary properties of integrals......Page 12
3 – Riemann sums. The integral notation......Page 21
4 – Uniform limits of integrable functions......Page 23
5 – Application to Fourier series and to power series......Page 28
6 – The Borel-Lebesgue Theorem......Page 33
7 – Integrability of regulated or continuous functions......Page 36
8 – Uniform continuity and its consequences......Page 38
9 – Differentiation and integration under the sign......Page 43
10 – Semicontinuous functions15......Page 48
11 – Integration of semicontinuous functions......Page 55
12 – The fundamental theorem of the differential and integral calculus......Page 59
13 – Extension of the fundamental theorem to regulated functions......Page 66
14 – Convex functions; Holder and Minkowski inequalities......Page 72
15 – Integration by parts......Page 81
16 – The square wave Fourier series......Page 84
17 – Wallis’ formula......Page 87
18 – Taylor’s Formula......Page 89
19 – Change of variable in an integral......Page 98
20 – Integration of rational fractions......Page 102
21 – Convergent integrals: examples and definitions......Page 109
22 – Absolutely convergent integrals......Page 111
23 – Passage to the limit under the sign......Page 116
24 – Series and integrals......Page 122
25 – Differentiation under the sign......Page 125
26 – Integration under the sign......Page 131
27 – How to make C∞ a function which is not......Page 136
28 – Approximation by polynomials......Page 142
29 – Functions having given derivatives at a point......Page 145
30 – Radon measures on a compact set......Page 148
31 – Measures on a locally compact set......Page 157
32 – The Stieltjes construction......Page 164
33 – Application to double integrals......Page 171
34 – Definition and examples......Page 175
35 – Derivatives of a distribution......Page 180
Appendix to Chapter V Introduction to the Lebesgue Theory......Page 185
1 – Comparison relations......Page 201
2 – Rules of calculation......Page 203
3 – Truncated expansions......Page 204
4 – Truncated expansion of a quotient......Page 206
5 – Gauss’ convergence criterion......Page 208
6 – The hypergeometric series......Page 210
7 – Asymptotic study of the equation xex = t......Page 212
8 – Asymptotics of the roots of sin x. log x = 1......Page 214
9 – Kepler’s equation......Page 216
10 – Asymptotics of the Bessel functions......Page 219
11 – Cavalieri and the sums 1k + 2k + . . . + nk......Page 230
12 – Jakob Bernoulli......Page 232
13 – The power series for cot z......Page 237
14 – Euler and the power series for arctan x......Page 240
15 – Euler, Maclaurin and their summation formula......Page 244
16 – The Euler-Maclaurin formula with remainder......Page 245
17 – Calculating an integral by the trapezoidal rule......Page 247
18 – The sum 1+1/2 + . . . + 1/n, the infinite product for the Γ function, and Stirling’s formula......Page 248
19 – Analytic continuation of the zeta function......Page 253
1 – Cauchy’s integral formula for a circle......Page 256
2 – Functions and measures on the unit circle......Page 260
3 – Fourier coefficients......Page 266
4 – Convolution product on T......Page 271
5 – Dirac sequences in T......Page 275
6 – Absolutely convergent Fourier series......Page 279
7 – Hilbertian calculations......Page 280
8 – The Parseval-Bessel equality......Page 282
9 – Fourier series of differentiable functions......Page 288
10 – Distributions on T......Page 292
11 – Dirichlet’s theorem......Page 300
12 – Fejer’s theorem......Page 306
13 – Uniformly convergent Fourier series......Page 308
§ 4. Analytic and holomorphic functions......Page 312
14 – Analyticity of the holomorphic functions......Page 313
15 – The maximum principle......Page 315
16 – Functions analytic in an annulus. Singular points. Meromorphic functions......Page 318
17 – Periodic holomorphic functions......Page 324
18 – The theorems of Liouville and of d’Alembert-Gauss......Page 325
19 – Limits of holomorphic functions......Page 335
20 – Infinite products of holomorphic functions......Page 337
21 – Analytic functions defined by a Cauchy integral......Page 345
22 – Poisson’s function......Page 347
23 – Applications to Fourier series......Page 349
24 – Harmonic functions......Page 352
25 – Limits of harmonic functions......Page 356
26 – The Dirichlet problem for a disc......Page 359
27 – The Poisson summation formula......Page 362
28 – Jacobi’s theta function......Page 366
29 – Fundamental formulae for the Fourier transform......Page 370
30 – Extensions of the inversion formula......Page 374
31 – The Fourier transform and differentiation......Page 379
32 – Tempered distributions......Page 383
§ 1. How to fool young innocents......Page 392
§ 2. The evolution of R&D funding in America......Page 413
§ 3. Applied mathematics in America......Page 429
Index......Page 441
Table of Contents of Volume I......Page 445
Universitext......Page 448
