| “天元基金影印数学丛书”主要包含国外反映近代数学发展的纯数学与应用数学方面的优秀书籍，天元基金邀请国内各个方向的知名数学家参与选题的工作，经专家遴选、推荐，由高等教育出版社影印出版。《分析》一书第一卷的内容包括集合与函数、离散变量的收敛性、连续变量的收敛性、幂函数、指数函数与三角函数；第二卷的内容包括Fourier级数和Fourier积分以及可以通过Fourier级数解释的Weierstrass的解析函数理论。《分析》可作为高年级本科生教材或参考书。 |
| 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. Taylors Formula 18 - Taylors 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 - Keplers 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 Stirlings formula 19 - Analytic continuation of the zeta function VII - Harmonic Analysis and Holomcrphic Functions 1 - Cauchys 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. Dirichlets method 11 - Dirichlets theorem 12 - Fejers 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 dAlembert-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 - Poissons 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 - Jacobis 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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