Ⅴ-Differential and Integral Calculus 1
1．The Riemann Integral 1
1-Upper and lower integrals of a bounded function 1
2-Elementary properties of integrals 5
3-Riemann sums.The integral notation 14
4-Uniform limits of integrable functions 16
5-Application to Fourier series and to power series 21
2．Integrability Conditions 26
6-The Borel-Lebesgue Theorem 26
7-Integrability of regulated or continuous functions 29
8-Uniform continuity and its consequences 31
9-Differentiation and integration under the ∫ sign 36
10-Semicontinuous functions 41
11-Integration of semicontinuous functions 48
3．The"Fundamental Theorem"(FT) 52
12-The fundamental theorem of the differential and integral calculus 52
13-Extension of the fundamental theorem to regulated func-tions 59
14-Convex functions;H？lder and Minkowski inequalities 65
4．Integration by parts 74
15-Integration by parts 74
16-The square wave Fourier series 77
17-Wallis'formula 80
5．Taylor's Formula 82
18-Taylor's Formula 82
6．The change of variable formula 91
19-Change of variable in an integral 91
20-Integration of rational fractions 95
7．Generalised Riemann integrals 102
21-Convergent integrals:examples and definitions 102
22-Absolutely convergent integrals 104
23-Passage to the limit under the ∫ sign 109
24-Series and integrals 115
25-Differentiation under the ∫ sign 118
26-Integration under the ∫ sign 124
8．Approximation Theorems 129
27-How to make C∞ a function which is not 129
28-Approximation by polynomials 135
29-Functions having given derivatives at a point 138
9．Radon measures in R or C 141
30-Radon measures on a compact set 141
31-Measures on a locally compact set 150
32-The Stieltjes construction 157
33-Application to double integrals 164
10．Schwartz distributions 168
34-Definition and examples 168
35-Derivatives of a distribution 173
Appendix to Chapter Ⅴ-Introduction to the Lebesgue Theory 179
Ⅵ-Asymptotic Analysis 195
1．Truncated expansions 195
1-Comparison relations 195
2-Rules of calculation 197
3-Truncated expansions 198
4-Truncated expansion of a quotient 200
5-Gauss'convergence criterion 202
6-The hypergeometric series 204
7-Asymptotic study of the equation xex=t 206
8-Asymptotics of the roots of sin x log x=1 208
9-Kepler's equation 210
10-Asymptotics of the Bessel functions 213
2．Summation formulae 224
11-Cavalieri and the sums 1k+2k+...+nk 224
12-Jakob Bernoulli 226
13-The power series for cot z 231
14-Euler and the power series for arctanx 234
15-Euler,Maclaurin and their summation formula 238
16-The Euler-Maclaurin formula with remainder 239
17-Calculating an integral by the trapezoidal rule 241
18-The sum 1+1/2+...+1/n,the infinite product for the Γ function,and Stirling's formula 242
19-Analytic continuation of the zeta function 247
Ⅶ-Harmonic Analysis and Holomorphic Functions 251
1-Cauchy's integral formula for a circle 251
1．Analysis on the unit circle 255
2-Functions and measures on the unit circle 255
3-Fourier coefficients 261
4-Convolution product on T 266
5-Dirac sequences in T 270
2．Elementary theorems on Fourier series 274
6-Absolutely convergent Fourier series 274
7-Hilbertian calculations 275
8-The Parseval-Bessel equality 277
9-Fourier series of difierentiable functions 283
10-Distributions on T 287
3．Dirichlet's method 295
11-Dirichlet's theorem 295
12-Fejér's theorem 301
13-Uniformly convergent Fourier series 303
4．Analytic and holomorphic functions 307
14-Analyticity of the holomorphic functions 308
15-The maximum principle 310
16-Functions analytic in an annulus.Singular points.Mero-morphic functions 313
17-Periodic holomorphic functions 319
18-The theorems of Liouville and of d'Alembert-Gauss 320
19-Limits of holomorphic functions 330
20-Infinite products of holomorphic functions 332
5．Harmonic functions and Fourier series 340
21-Analytic functions defined by a Cauchy integral 340
22-Poisson's function 342
23-Applications to Fourier series 344
24-Harmonic functions 347
25-Limits of harmonic functions 351
26-The Dirichlet problem for a disc 354
6．From Fourier series to integrals 357
27-The Poisson summation formula 357
28-Jacobi's theta function 361
29-Fundamental formulae for the Fourier transform 365
30-Extensions of the inversion formula 369
31-The Fourier transform and differentiation 374
32-Tempered distributions 378
Postface.Science,technology,arms 387
Index 436
Table of Contents of Volume Ⅰ 441