## A Course on Mathematical Analysis – Khinchin

In this post, we will see the book A Course on Mathematical Analysis by  A. Ya. Khinchin. This course of mathematical analysis is a text-book for students of mechanico-mathematicaland physico-mathematical faculties of our universities (and to some extent of pedagogical institutes as well) ; it is intended as the main text-book in the study of a science which appears in the curriculum under the heading of mathematical analysis and which deals with the theory of limits, infinite series and differential calculus with simple applications of these subjects.

The book was translated from Russian was published in 1957.

You can get the book here.

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# Contents

## Chapter 1. FUNCTIONS 1

§ 1. Variables 1
§ 2. Functions 3
§ 3. The region of definition of a function 6
§ 4. Functions and formulae 7
§ 5. The geometrical representation of functions 11
§ 6. Elementary functions 13

## Chapter 2. ELEMENTARY THEORY OF LIMITS 18

§ 7. Infinitesimal quantities 18
§ 8. Operations with infinitesimal quantities 23
§ 9. Infinitely large quantities 26
§ 10. Quantities which tend to limits 29
§ 11. Operations with quantities which tend to limits 33
§ 12. Infinitesimal and infinitely large quantities of different orders 39

## Chapter 3. THE DEVELOPMENT OF THE ACCURATE THEORY OF LIMIT TRANSITION 45

§ 13. The mathematical definition of a process 45
§ 14. The accurate concept of limits 47
§ 15. The development of the concept of limit transitions 52

## Chapter 4. REAL NUMBERS 56

§ 16. Necessity of producing a general theory of real numbers 56
§ 17. Construction of a continuum 59
§ 18. Fundamental lemmas 69
§ 19. Final points in connection with the theory of limits 74

## Chapter 5. CONTINUOUS FUNCTIONS 79

§ 20. Definition of continuity 79
§ 21. Operations with continuous functions 84
§ 22. Continuity of a composite function 85
§ 23. Fundamental properties of continuous functions 87
§ 24. Continuity of elementary functions 94

## Chapter 6. DERIVATIVES 98

§ 25. Uniform and non-uniform variation of functions 98
§ 26. Instantaneous velocity of non-uniform movement 101
§ 27. Local density of a heterogeneous rod 106
§ 28. Definition of a derivative 108
§ 29. Laws of differentiation 110
§ 30. The existence of functions and their geometrical illustration 123

## Chapter 7. DIFFERENTIALS 128

§ 31. Definition and relationship with derivatives 128
§ 32. Geometrical illustration and laws for evaluation 132
§ 33. Invariant character of the relationship between a derivative and a differential 134

## Chapter 8. DERIVATIVES AND DIFFERENTIALS OF HIGHER ORDERS 136

§ 34. Derivatives of higher orders 136
§ 35. Differentials of higher orders and their relationship with derivatives 139

## Chapter 9. MEAN VALUE THEOREMS 142

§ 36. Theorem on finite increments 142
§ 37. Evaluation of limits of ratios of infinitely small and infinitely large quantities 147
§ 38. Taylor’s formula 154
§ 39. The last term in Taylor’s formula 158

## Chapter 10. APPLICATION OF DIFFERENTIAL CALCULUS TO ANALYSIS OF FUNCTIONS 164

§ 40. Increasing and decreasing of functions 164
§ 4l. Extrema 167

## Chapter 11. INVERSE OF DIFFERENTIATION 175

§ 42. Concept of primitives 175
§ 43. Simple general methods of integration 182

## Chapter 12. INTEGRAL 193

§ 44. Area of a curvilinear trapezium 193
§ 45. Work of a variable force 198
§ 46. General concept of an integral 201
§ 47. Upper and lower sums 204
§ 48. Integreability of functions 207

## Chapter 13. RELATIONSHIP BETWEEN AN INTEGRAL AND A PRIMITIVE 213

§ 49. Simple properties of integrals 213
§ 50. Relationship between an integral and a primitive 218
§ 51. Further properties of integrals 223

## Chapter 14. THE GEOMETRICAL AND MECHANICAL APPLICATIONS OF INTEGRALS 230

§ 52. Length of an arc of a plane curve 230
§ 53. Lengths of arcs of curves in space 241
§ 54. Mass, centre of gravity and moments of inertia of a material plane curve 242
§ 55. Capacities of geometrical bodies 247

## Chapter 15. APPROXIMATE EVALUATION OF INTEGRALS 254

§ 56. Problematic set up 254
§ 57. Method of trapeziums 257
§ 58. Method of parabolas 262

## Chapter 16. INTEGRATION OF RATIONAL FUNCTIONS 265

§ 59. Algebraical introduction 265
§ 60. Integration of simple fractions 274

## Chapter 17. INTEGRATION OF THE SIMPLE RATIONAL AND TRANSCENDENTAL FUNCTIONS 282

§ 62. Integration of functions of the type R(x,\sqrt[n]{\frac{ax+b}{cx+d}}) 282
§ 63. Integration of functions of the type R (x, \sqrt{ax^{2}+ bx +c}) 284
§ 64. Primitives of binomial differentials 287
§ 65. Integration of trigonometrical differentials 289
§ 66. Integration of differentials containing exponential
functions 294

## Chapter 18. NUMERICAL INFINITE SERIES 297

§ 67. Fundamental concepts 297
§ 68. Series with constant signs 305
§ 69. Series with variable signs 316
§ 70. Operations with series 320
§ 71. Infinite products 326

## Chapter 19. INFINITE SERIES OF FUNCTIONS 333

§ 72. Region of convergence of a series of functions 333
§ 73. Uniform convergence 335
§ 74. The continuity of the sum of a functional series 340
§ 75. Term-by-term integration and differentiation of series 344

## Chapter 20 POWER SERIES AND SERIES OF POLYNOMIALS 351

§ 76. Region of convergence of a power series 351
§ 77. Uniform convergence and its consequences 357
§ 78. Expansion of functions into power series 361
§ 79. Series of polynomials 369
§ 80. Theorem of Weierstrass 372

## Chapter 21. TRIGONOMETRICAL SERIES 377

§ 81. Fourier coefficients 377
§ 82. Average approximation 383
§ 83. Dirichlet-Liapunov theorem on closed trigonometrical systems 388
§ 84. Convergence of Fourier series 394
§ 85. Generalised trigonometrical series 396

## Chapter 22. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES 400

§ 86. Continuity of functions of several independent variables 400
§ 87. Two-dimensional continuum 403
§ 88. Properties of continuous functions 408
§ 89. Partial derivatives 410
§ 90. Differentials 413
§ 91. Derivatives in arbitrary directions 419
§ 92. Differentiation of composite and implicit functions 422
§ 93. Homogeneous functions and Euler theorem 427
§ 94. Partial derivatives of higher orders 429
§ 95. Taylor’s formula for functions of two variables 433
§ 96. Extrema 438

## Chapter 23. SOME SIMPLE GEOMETRICAL APPLICATIONS OF DIFFERENTIAL CALCULUS 443

§ 97. Equations of tangent and normal to a plane curve 443
§ 98. Tangential line and normal plane to a curve in space 446
§ 99. Tangential and normal planes to a surface 448
§ 100. Direction of convexity and concavity of a curve 451
§ 101. Curvature of a plane curve 453
§ 102. Tangential circle 458

## Chapter 24. IMPLICIT FUNCTIONS 462

§ 103. The simplest problem 462
§ 104. The general problem 469
§ 106. Conditional extremum 483

## Chapter 25. GENERALISED INTEGRALS 491

§ 107. Integrals with infinite limits 491
§ 108. Integrals of unbounded functions 504

## Chapter 26. INTEGRALS OF PARAMETRIC FUNCTIONS 514

§ 109. Integrals with finite limits 514
§ 110. Integrals with infinite limits 526
§ 111. Examples 535
§ 112. Euler’s integrals 541
§ 113. Stirling’s formula 548

## Chapter 27. DOUBLE AND TRIPLE INTEGRALS 557

§ 114. Measurable plane figures 557
§ 115. Volumes of cylindrical bodies 567
§ 116. Double integral 571
§ 117. Evaluation of double integrals by means of two simple integrations 576
§ 118. Substitution of variables in double integrals 584
§ 119. Triple integrals 590
§ 120. Applications 593

## Chapter 28. CURVILINEAR INTEGRALS 602

§ 121. Definition of a plane curvilinear integral 602
§ 122. Wark of a plane field of force 610
§ 123. Green’s formula 612
§ 124, Application to differentials of functions of two variables 617
§ 125. Curvilinear integrals in space 622

## Chapter 29. SURFACE INTEGRALS 626

§ 126. The simplest case 626
§ 127. General definition of surface integrals 630
§ 129. Stoke’s formula 642
§ 130. Elements of the field theory 647

## CONCLUSION – Short historical sketch 653

INDEX 665 