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 Theory179
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 xe
x = 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 1
k + 2
k + . . . + n
k . . . . . . . . . . . . . 224
12 – Jakob Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
13 – The power series for cot z . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
14 – Euler and the power series for arctan x . . . . . . . . . . . . . . . . 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
320
Table of Contents of Volume II
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 Theory179
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 xe
x = 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 1
k + 2
k + . . . + n
k . . . . . . . . . . . . . 224
12 – Jakob Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
13 – The power series for cot z . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
14 – Euler and the power series for arctan x . . . . . . . . . . . . . . . . 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
320
Table of Contents of Volume II