V - Differential and Integral Calculus 1. The Riemann Integral 1 - Upper and lower integrals of a bounded function 2 - Elementary properties of integrals 3 - Riemann sums. The integral notation 4 - Uniform limits of integrable functions 5 - Application to Fourier series and to power series 2. Integrability Conditions 6 - The Borel-Lebesgue Theorem 7 - Integrability of regulated or continuous functions 8 - Uniform continuity and its consequences 9 - Differentiation and integration under the f sign 10 - Semicontinuous functions 11 - Integration of semicontinuous functions 3. The "Fundamental Theorem" （FT） 12 - The fundamental theorem of the differential and integral calculus 13 - Extension of the fundamental theorem to regulated functions 14 - Convex functions; Holder and Minkowski inequalities 4. Integration by parts 15 - Integration by parts 16 - The square wave Fourier series 17- Wallis' formula 5. Taylor's Formula 18 - Taylor's Formula 6. The change of variable formula 19 - Change of variable in an integral 20 - Integration of rational fractions 7. Generalised Riemann integrals 21 - Convergent integrals: examples and definitions 22 - Absolutely convergent integrals 23 - Passage to the limit under the f sign 24 - Series and integrals 25 - Differentiation under the f sign 26 - Integration under the f sign 8. Approximation Theorems 27 - How to make C a function which is not 28 - Approximation by polynomials 29 - Functions having given derivatives at a point 9. Radon measures in R or C 30 - Radon measures on a compact set 31 - Measures on a locally compact set 32 - The Stieltjes construction 33 - Application to double integrals 10. Schwartz distributions 34 - Definition and examples 35 - Derivatives of a distribution Appendix to Chapter V - Introduction to the Lebesgue Theory VI - Asymptotic Analysis 1. Truncated expansions 1 - Comparison relations 2 - Rules of calculation 3 - Truncated expansions 4 - Truncated expansion of a quotient 5 - Gauss' convergence criterion 6 - The hypergeometric series 7 - Asymptotic study of the equation xex = t 8 - Asymptotics of the roots of sin x log x = 1 9 - Kepler's equation 10 - Asymptotics of the Bessel functions 2. Summation formulae 11 - Cavalieri and the sums 1k + 2k + ... + nk 12 - Jakob Bernoulli 13 - The power series for cot z 14 - Euler and the power series for arctan x 15 - Euler, Maclaurin and their summation formula 16 - The Euler-Maclaurin formula with remainder 17 - Calculating an integral by the trapezoidal rule 18 - The sum 1 + 1/2 ... + l/n, the infinite product for the F function, and Stirling's formula 19 - Analytic continuation of the zeta function VII - Harmonic Analysis and Holomcrphic Functions 1 - Cauchy's integral formula for a circle 1. Analysis on the unit circle 2 - Functions and measures on the unit circle 3 - Fourier coefficients 4 - Convolution product on 5 - Dirac sequences in T 2. Elementary theorems on Fourier series 6 - Absolutely convergent Fourier series 7 - Hilbertian calculations 8 - The Parseval-Bessel equality 9 - Fourier series of differentiable functions 10 - Distributions on 3. Dirichlet's method 11 - Dirichlet's theorem 12 - Fejer's theorem 13 - Uniformly convergent Fourier series 4. Analytic and holomorphic functions 14 - Analyticity of the holomorphic functions 15 - The maximum principle 16 - Functions analytic in an annulus. Singular points. Meromorphic functions 17 - Periodic holomorphic functions 18 - The theorems of Liouville and d'Alembert-Gauss 19 - Limits of holomorphic functions 20 - Infinite products of holomorphic functions 5. Harmonic functions and Fourier series 21 - Analytic functions defined by a Cauchy integral 22 - Poisson's function 23 - Applications to Fourier series 24 - Harmonic functions 25 - Limits of harmonic functions 26 - The Dirichlet problem for a disc 6. From Fourier series to integrals 27 - The Poisson summation formula 28 - Jacobi's theta function 29 - Fundamental formulae for the Fourier transform 30 - Extensions of the inversion formula 31 - The Fourier transform and differentiation 32 - Tempered distributions Postface. Science, technology, arms Index Table of Contents of Volume I
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