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纯数学教程(纪念版)(英文版)

  • 定价: ¥99
  • ISBN:9787519253622
  • 开 本:16开 平装
  • 作者:(英)G.H.哈代
  • 立即节省:
  • 2019-09-01 第1版
  • 2019-09-01 第1次印刷
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导语

  

内容提要

  

    《纯数学教程(纪念版)》是“剑桥数学图书馆”系列丛书之一。这部部世纪经典著作,以简洁易懂的数学语言,全面系统地介绍了基础数学的各个方面,并对许多经典的数学论证给出了严谨的证明。本书共分10章,在介绍了实数、复数的概念后,从第4章和第5章引入了极限的概念,较之一般书的处理方法更为轻松自然、易于接受。另外,书中每章后面配有大量有代表性的杂例,供读者参考练习以巩固所学知识。本书适合高校数学系及对相关专业学生和教师学习和参考。

作者简介

    哈代(Hardy,Godfrey Harold,1877年2月7日-1947年12月1日),卒于剑桥 。13岁进入以培养数学家著称的温切斯特学院。1896年去剑桥三一学院,并于1900年在剑桥获得一个职位。同年得史密斯奖。以后,在英国牛津大学、剑桥大学任教授。他和J.E.李特尔伍德长期进行合作,写出了近百篇论文,在丢番图逼近,堆垒数论、黎曼ξ函数、三角级数、不等式、级数与积分等领域作出了很大贡献,同时是回归数现象发现者。在20世纪上半叶建立了具有世界水平的英国分析学派。

目录

CHAPTER Ⅰ  REAL VARIABLES
  1-2.Rational numbers
  3-7.Irrational numbers
  8.Real numbers
  9.Relations of magnitude between real numbers
  10-11.Algebraical operations with real numbers
  12.The number√2
  13-14.Quadratic surds
  15.The continuum
  16.The continuous real variable
  17.Sections of the real numbers. Dedekind's theorem
  18.Points of accumulation
  19.Weierstrass's theorem
  Miscellaneous examples
CHAPTER Ⅱ  FUNCTIONS OF REAL VARIABLES
  20.The idea of a function
  21.The graphical representation of functions. Coordinates
  22.Polar coordinates
  23.Polynomia s
  24-25.Rational functions
  26-27.Algebraical functions
  28-29.Transcendental functions
  30.Graphical solution of equations
  31.Functions of two variables and their graphical representation
  32.Curves in a plane
  33.Loci in space
  Miscellaneous examples
CHAPTER Ⅲ  COMPLEX NUMBER
  34-38.Displacements
  39-42.Complex numbers
  43.The quadratic equation with real coefficients
  44.Argand's diagram
  45.De Moivre's theorem
  46.Rational functions of a complex variable
  47-49.Roots of complex numbers
  Miscellaneous examples
CHAPTER Ⅳ  LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
  50.Functions of a positive integral variable
  51.Interpolation
  52.Finite and infinite classes
  53-57.Properties possessed by a function of n for large values of n
  58-61.Definition of a limit and other definitions
  62.Oscillating functions
  63-68.General theorems concerning limits
  69-70.Steadily increasing or decreasing functions
  71.Alternative proof of Weierstrass's theorem
  72.The limit of xn
  73.The limit of (1+1/n) n
  74.Some algebraical lemmas
  75.The limit of n □
  76-77.Infinite series
  78.The infinite geometrical series
  79.The representation of functions of a continuous real variable by means of limits
  80.The bounds of a bounded aggregate
  81.The bounds of a bounded function
  82.The limits of indetermination of a bounded function
  83-84.The general principle of convergence
  85-86.Limits of complex functions and series of complex terms
  87-88.Applications to zn and the geometrical series
  89.The symbols 0, o, ~
  Miscellaneous examples
CHAPTER Ⅴ  LIMITSOFFUNCTIONSOFACONTINUOUSVARIABLE.CONTINUOUS AND DISCONTINUOUS FUNCTIONS
  90-92.Limits as x→ ∞ or x → ∞
  93-97.Limits as x → a
  98.The symbols O, o, ~: orders of smallness and greatness
  99-100.Continuous functions of a real variable
  101-105.Properties of continuous functions. Bounded functions The oscillation of a function in an interval
  106-107.Sets of intervals on a line. The Heine-Borel theorem
  108.Continuous functions of several variables
  109-110.Implicit and inverse functions
  Miscellaneous examples
CHAPTER Ⅵ  DERIVATIVES AND INTEGRALS
  111-113.Derivatives
  114.General rules for diferentiation
  115.Derivatives of complex functions
  116.The notation of the differential calculus
  117.Differentiation of polynomials
  118.Differentiation of rational functions
  119.Differentiation of algebraical functions
  120.Differentiation of transcendental functions
  121.Repeated differentiation
  122.General theorems concerning derivatives Rolle's theorem
  123-125.Maxima and minima
  126-127.The mean value theorem
  128.Cauchy's mean value theorem
  129.A theorem of Darboux
  130-131.Integration. The logarithmic function
  132.Integration of polynomials
  133-134.Integration of rational functions
  135-142.Integration of algebraical functions. Integration by rationalisation. Integration by parts
  143-147. Integration of transcendental functions
  148.Areas of plane curves
  149.Lengths of plane curves
  Miscellaneous examples
CHAPTER Ⅶ  ADDTTTONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL CALCULUS
  150-151.Taylor's theorem
  152.Taylor's series
  153.Applications of Taylor's theorem to maxima and minima
  154.The calculation of certain limits
  155.The contact of plane curves
  156-158. Differentiation of functions of several variables
  159.The mean value theorem for functions of two variables
  160.Differentials
  161-162.Definite integrals
  163.The circular functions
  164.Calculation of the definite integral as the limit of a sum
  165.General properties of the definite integral
  166.Integration by parts and by substitution
  167.Alternative proof of Taylor's theorem
  168.Application to the binomial series
  169.Approximate formulae for definite integrals. Simpson's rule
  170.Integrals of complex functions
  Miscellaneous examples
CHAPTER Ⅷ  THE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALS
  171-174.Series of positive terms.Cauchy's and d'Alembert's tests of convergence
  175.Ratio tests
  176.Dirichlet's theorem
  177.Multiplication of series of positive terms
  178-180.Further tests for convergence. Abel's theorem. Maclaurln's integral test
  181.The series ∑n-3
  182.Cauchy's condensation test
  183.Further ratio tests
  184-189.Infinite integrals
  190.Series of positive and negative terms
  191-192.Absolutely convergent series
  193-194.Conditionally convergent series
  195.Alternating series
  196.Abel's and Dirichlet's tests of convergence
  197.Series of complex terms
  198-201.Power series
  202.Multiplication of series
  203.Absolutely and conditionally convergent infinite integrals
  Miscellaneous examples
CHAPTER Ⅸ  THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS OF A REAL VARIABLE
  204-205.The logarithmic function
  206.The functional equation satisfied by log x
  207-209.The behaviour of log x as x tends to infinity or to zero
  210.The logarithmic scale of infinity
  211.The number e
  212-213.The exponential function
  214.The general power ax
  215.The exponential limit
  216.The logarithmic limit
  217.Common logarithms
  218.Logarithmic tests of convergence
  219.The exponential series
  220.The logarithmic series
  221.The series for arc tan x
  222.The binomial series
  223.Alternative development of the theory
  224-226.The analytical theory of the circular functions
  Miscellaneous examples
CHAPTER Ⅹ  THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
  227-228.Functions of a complex variable
  229.Curvilincar integrals
  230.Definition of the logarithmic function
  231.The values of the logarithmic function
  232-234.The exponential function
  235-236.The general power aζ
  237-240.The trigonometrical and hyperbolic functions
  241.The connection between the logarithmic and inverse trigonometrical functions
  242.The exponential series
  243.The series for cos z and sin z
  244-245.The logarithmic series
  246.The exponential limit
  247.The binomial series
  Miscellaneous examples
APPENDIX Ⅰ  The proof that every equation has a root
APPENDIX Ⅱ  A note on double limit problems
APPENDIX Ⅲ  The infinite in analysis and geometry
APPENDIX Ⅳ  The infinite in analysis and geometry
INDEX