Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Functions

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