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Ⅴ-Differential and Integral Calculus1

1．The Riemann Integral1

1-Upper and lower integrals of a bounded function1

2-Elementary properties of integrals5

3-Riemann sums.The integral notation14

4-Uniform limits of integrable functions16

5-Application to Fourier series and to power series21

2．Integrability Conditions26

6-The Borel-Lebesgue Theorem26

7-Integrability of regulated or continuous functions29

8-Uniform continuity and its consequences31

9-Differentiation and integration under the ∫ sign36

10-Semicontinuous functions41

11-Integration of semicontinuous functions48

3．The"Fundamental Theorem"(FT)52

12-The fundamental theorem of the differential and integral calculus52

13-Extension of the fundamental theorem to regulated func-tions59

14-Convex functions;H？lder and Minkowski inequalities65

4．Integration by parts74

15-Integration by parts74

16-The square wave Fourier series77

17-Wallis'formula80

5．Taylor's Formula82

18-Taylor's Formula82

6．The change of variable formula91

19-Change of variable in an integral91

20-Integration of rational fractions95

7．Generalised Riemann integrals102

21-Convergent integrals:examples and definitions102

22-Absolutely convergent integrals104

23-Passage to the limit under the ∫ sign109

24-Series and integrals115

25-Differentiation under the ∫ sign118

26-Integration under the ∫ sign124

8．Approximation Theorems129

27-How to make C∞ a function which is not129

28-Approximation by polynomials135

29-Functions having given derivatives at a point138

9．Radon measures in R or C141

30-Radon measures on a compact set141

31-Measures on a locally compact set150

32-The Stieltjes construction157

33-Application to double integrals164

10．Schwartz distributions168

34-Definition and examples168

35-Derivatives of a distribution173

Appendix to Chapter Ⅴ-Introduction to the Lebesgue Theory179

Ⅵ-Asymptotic Analysis195

1．Truncated expansions195

1-Comparison relations195

2-Rules of calculation197

3-Truncated expansions198

4-Truncated expansion of a quotient200

5-Gauss'convergence criterion202

6-The hypergeometric series204

7-Asymptotic study of the equation xex=t206

8-Asymptotics of the roots of sin x log x=1208

9-Kepler's equation210

10-Asymptotics of the Bessel functions213

2．Summation formulae224

11-Cavalieri and the sums 1k+2k+...+nk224

12-Jakob Bernoulli226

13-The power series for cot z231

14-Euler and the power series for arctanx234

15-Euler,Maclaurin and their summation formula238

16-The Euler-Maclaurin formula with remainder239

17-Calculating an integral by the trapezoidal rule241

18-The sum 1+1/2+...+1/n,the infinite product for the Γ function,and Stirling's formula242

19-Analytic continuation of the zeta function247

Ⅶ-Harmonic Analysis and Holomorphic Functions251

1-Cauchy's integral formula for a circle251

1．Analysis on the unit circle255

2-Functions and measures on the unit circle255

3-Fourier coefficients261

4-Convolution product on T266

5-Dirac sequences in T270

2．Elementary theorems on Fourier series274

6-Absolutely convergent Fourier series274

7-Hilbertian calculations275

8-The Parseval-Bessel equality277

9-Fourier series of difierentiable functions283

10-Distributions on T287

3．Dirichlet's method295

11-Dirichlet's theorem295

12-Fejér's theorem301

13-Uniformly convergent Fourier series303

4．Analytic and holomorphic functions307

14-Analyticity of the holomorphic functions308

15-The maximum principle310

16-Functions analytic in an annulus.Singular points.Mero-morphic functions313

17-Periodic holomorphic functions319

18-The theorems of Liouville and of d'Alembert-Gauss320

19-Limits of holomorphic functions330

20-Infinite products of holomorphic functions332

5．Harmonic functions and Fourier series340

21-Analytic functions defined by a Cauchy integral340

22-Poisson's function342

23-Applications to Fourier series344

24-Harmonic functions347

25-Limits of harmonic functions351

26-The Dirichlet problem for a disc354

6．From Fourier series to integrals357

27-The Poisson summation formula357

28-Jacobi's theta function361

29-Fundamental formulae for the Fourier transform365

30-Extensions of the inversion formula369

31-The Fourier transform and differentiation374

32-Tempered distributions378

Postface.Science,technology,arms387

Index436

Table of Contents of Volume Ⅰ441