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 functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
14 – Convex functions; H¨ older 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
Table of Contents of Volume II
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