In this post, we will see the book Eight Lectures On Mathematical Analysis by A. Ya. Khinchin.

# About the book

Eight Lectures on Mathematical Analysis is a translation and adaptation of a book by the outstanding Russian mathematician A. Ya. Khinchin. It is based on a series of lectures delivered at the University of Moscow by Professor Khinchin to improve the mathematical qualifications of engineers.In this book, the reader will find a masterful outline of the fundamental ideas of mathematical analysis. Inessential details have purposely been omitted, and the resulting exposition is clear and easy to follow. The book should be accessible to anyone who has had even a sketchy introduction to the material. And yet, because it is a concise, lucid exposition of the most important concepts of mathematical analysis, the book should be of value to the student enrolled in a university course in analysis.A. YA. KHINCHIN, until his death in 1959, was a professor at Moscow State University, a corresponding member of the Academy of Sciences, and a member of the Academy of Pedagogical Sciences of the RSFSR. The author of more than one hundred fifty mathematical research papers and books, he will be remembered as a world-renowned authority in mathematical analysis, probability theory, number theory, and mathematical statistics.

The book was translated from Russian by Irena Zygmud was published in 1965.

Credits to original uploader.

*Note:* This is one of the clearest exposition of these fundamental mathematical concepts that I have come across. Khinchin was a genius mathematician teacher who develops the concepts very gradually not assuming much and explains subtle points in the process which are usually missed out. Also, we never lose sight of what we are trying to achieve in a derivation with a clear logical path towards it and its consequences for the discussion. I hope you enjoy this book as much as I did.

You can get the book here.

Follow us on The Internet Archive: https://archive.org/details/@mirtitles

Follow Us On Twitter: https://twitter.com/MirTitles

Write to us: mirtitles@gmail.com

Fork us at GitLab: https://gitlab.com/mirtitles/

Add new entries to the detailed book catalog here.

# Contents

## LECTURE 1. The Continuum 1

1. Why begin with the continuum? 1

2. Need for a theory of real numbers 3

3. Construction of the irrational numbers 7

4. Theory of the continuum 11

5. Fundamental lemmas of the real number system 16

## LECTURE 2. Limits 22

6. What is a limit? 22

7. Some ways of tending toward a limit 24

8. The limit of a constant function 27

9. Infinitely small and infinitely large quantities 28

10. Cauchy’s condition for the limit of a function 31

11. A remark on the fundamental theorems on limits 33

12. Partial limits; the upper and lower limits 33

13. Limits of functions of several variables 40

## LECTURE 3. Functions 44

14. What is a function? 44

15. The domain of a function 49

16. Continuity of a function 50

17. Bounded functions 52

18. Basic properties of continuous functions 55

19. Continuity of the elementary functions 60

20. Oscillation of a function at a point 63

21. Points of discontinuity 65

22. Monotonic functions 67

23. Functions of bounded variation 69

## LECTURE 4. Series 71

24. Convergence and the sum of a series 71

25. Cauchy’s condition for convergence 74

26. Series with positive terms 75

27. Absolute and conditional convergence 81

28. Infinite products 84

29. Series of functions 88

30. Power Series 96

## LECTURE 5. The Derivative 102

31. The derivative and derivates 102

32. The differential 108

33. Lagrange’s theorem (first mean value theorem) 113

34. Derivatives and differentials of higher order 116

35. Limits of ratios of infinitely small and infinitely large quantities 118

36. Taylor’s formula 121

37. Maxima and minima 125

38. Partial derivatives 127

39. Differentiating implicit functions 132

## LECTURE 6. The Integral 137

40. Introduction 137

41. Definition of the integral 138

42. Criteria for integrability 144

43. Geometric and physical applications 148

44. Relation of integration to differentiation 152

45. Mean value theorems for integrals 154

46. Improper integrals 158

47. Double integrals 164

48. Evaluation of double integrals 169

49. The general operation of integration 173

## LECTURE 7. Expansion of functions in series 177

50. Use of series in the study of functions 177

51. Expansion in power series 179

52. Series of polynomials and the Weierstrass theorem 183

53. Trigonometric series 190

54. Fourier coefficients 192

55. Approximation in the mean 194

56. Completeness of the system of trigonometric functions 197

57. Convergence of Fourier series for functions with a bounded integrable derivative 201

58. Extension to arbitrary intervals 203

## LECTURE 8. Differential Equations 206

59. Fundamental concepts 206

60. The existence of a solution 211

61. Uniqueness of the solution 220

62. Dependence of the solution on parameters 222

63. Change of variables 226

64. Systems of equations of higher orders 230

Pingback: Eight Lectures on Mathematical Analysis – Khinchin — Mir Books | Chet Aero Marine