Eight Lectures on Mathematical Analysis – Khinchin

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.

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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

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1 Response to Eight Lectures on Mathematical Analysis – Khinchin

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

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