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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