# A Course of Mathematical Analysis

### International Series of Monographs on Pure and Applied Mathematics

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A Course of Mathematical Analysis, Part I is a textbook that shows the procedure for carrying out the various operations of mathematical analysis. Propositions are given with a precise statement of the conditions in which they hold, along with complete proofs. Topics covered include the concept of function and methods of specifying functions, as well as limits, derivatives, and differentials. Definite and indefinite integrals, curves, and numerical, functional, and power series are also discussed. This book is comprised of nine chapters and begins with an overview of mathematical analysis and its meaning, together with some historical notes and the geometrical interpretation of numbers. The reader is then introduced to functions and methods of specifying them; notation for and classification of functions; and elementary investigation of functions. Subsequent chapters focus on limits and rules for passage to the limit; the concepts of derivatives and differentials in differential calculus; definite and indefinite integrals and applications of integrals; and numerical, functional, and power series. This monograph will be a valuable resource for engineers, mathematicians, and students of engineering and mathematics.

## Table of Contents

Preface to the Seventh Edition

Introduction

1. Mathematical Analysis and Its Meaning

1. "Elementary" and "Higher" Mathematics

2. Magnitudes. Variables and Functional Relationships

3. Mathematical Analysis and Reality

2. Some Historical Notes

4. Great Russian Mathematicians: L. P. Euler, N. I . Lobachevskii, P. L. Chebyshev

5. Leading Russian Applied Mathematicians: N. E. Zhukovskii, S. A. Chaplygin, A. N. Krylov

3. Real Numbers

6. Real Numbers. The Real Axis

7. Intervals. Absolute Values

8. a Note on Approximations

Chapter I Functions

1. Functions and Methods of Specifying Them

9. The Concept of Function

10. Methods of Specifying Functions

2. Notation for and Classification of Functions

11. Notation

12. Function of a Function. Elementary Functions

13. The Classification of Functions

3. Elementary Investigation of Functions

14. Domain of Definition of a Function. Domain of Definiteness of an Analytic Expression

15. Elements of the Behavior of Functions

16. Graphical Investigation of a Function. Linear Combinations of Functions

4. Elementary Functions

17. Direct Proportionality and Linear Functions. Increments

18. Quadratic Functions

19. Inverse Proportionality and Linear Rational Functions

5. Inverse Functions. Power, Exponential and Logarithmic Functions

20. The Concept of Inverse Function

21. Power Functions

22. Exponential and Hyperbolic Functions

23. Logarithmic Functions

6. Trigonometric and Inverse Trigonometric Functions

24. Trigonometric Functions

25. Simple and Compound Harmonic Vibrations

26. Inverse Trigonometric Functions

Chapter II Limits

1. Basic Definitions

27. The Limit of a Function of an Integral Argument

28. Examples

29. The Limit of a Function of a Continuous Argument

2. Non-Finite Magnitudes. Rules for Passage to the Limit

30. Infinitely Large Magnitudes. Bounded Functions

31. Infinitesimals

32. Rules for Passage to the Limit

33. Examples

34. Tests for the Existence of a Limit

3. Continuous Functions

35. Continuity of a Function

36. Points of Discontinuity of a Function

37. General Properties of Continuous Functions

38. Operations on Continuous Functions. Continuity of the Elementary Functions

4. Comparison of Infinitesimals. Some Important Limits

39. Comparison of Infinitesimals. Equivalent Infinitesimals

40. Examples of Ratios of Infinitesimals

41. The Number e. Natural Logarithms

Chapter III Derivatives and Differentials. The Differential Calculus

1. The Concept of Derivative. Rate of Change of a Function

42. Some Physical Concepts

43. Derivative of a Function

44. Geometrical Interpretation of Derivative

45. Some Properties of the Parabola

2. Differentiation of Functions

46. Differentiation of the Results of Arithmetical Operations

47. Differentiation of a Function of a Function

48. Derivatives of the Basic Elementary Functions

49. Logarithmic Differentiation. Differentiation of Inverse and Implicit Functions

50. Graphical Differentiation

3. Differentials. Differentiability of a Function

51. Differentials and Their Geometrical Interpretation

52. Properties of the Differential

53. Application of the Differential to Approximations

54. Differentiability of a Function. Smoothness of a Curve

4. Derivative as Rate of Change (Further Examples)

55. Rate of Change of a Function with Respect to a Function. Parametric Specification of Functions and Curves

56. Rate of Change of Radius Vector

57. Rate of Change of Length of Arc

58. Processes of Organic Growth

5. Repeated Differentiation

59. Derivatives of Higher Orders

60. Leibniz's Formula

61. Differentials of Higher Orders

Chapter IV The Investigation of Functions and Curves

1. The Behavior of a Function "at a Point"

62. Construction of a Graph from "Elements" 197

63. Behavior of a Function "at a Point". Extrema

64. Tests for the Behavior of a Function "at a Point"

2. Applications of the First Derivative

65. Theorems of Rolle and Lagrange

66. Application of Lagrange's Formula to Approximations

67. Behavior of a Function in an Interval

68. Examples

69. a Property of the Primitive

3. Applications of the Second Derivative

70. Second Sufficient Test for an Extremum

71. Convexity and Concavity of a Curve. Points of Inflexion

72. Examples

4. Auxiliary Problems. Solution of Equations

73. Cauchy's Theorem and L'Hôpital's Rule

74. Asymptotic Variation of Functions and the Asymptotes of Curves

75. General Scheme for Investigation of Functions. Examples

76. Solution of Equations. Multiple Roots

5. Taylor's Formula and Its Applications

77. Taylor's Formula for Polynomials

78. Taylor's Formula

79. Some Applications of Taylor's Formula. Examples

6. Curvature

80. Curvature

81. Radius, Center and Circle of Curvature

82. Evolute and Involute

83. Examples

Chapter V The Definite Integral

1. The Definite Integral

84. Area of a Curvilinear Trapezoid

85. Examples From Physics

86. The Definite Integral. Existence Theorem

87. Evaluation of the Definite Integral

2. Basic Properties of the Definite Integral

88. Elementary Properties of the Definite Integral

89. Change of Direction and Subdivision of the Interval of Integration. Geometrical Interpretation of the Integral

90. Estimation of the Definite Integral

3. Basic Properties of the Definite Integral (Continued). The Newton-Leibniz Formula

91. Mean Value Theorem. Mean Value of a Function

92. Derivative of an Integral with Respect to Its Upper Limit

93. The Newton-Leibniz Formula

Chapter VI The Indefinite Integral. The Integral Calculus

1. The Indefinite Integral and Indefinite Integration

94. The Indefinite Integral. Basic Table of Integrals

95. Elementary Rules for Integration

96. Examples

2. Basic Methods of Integration

97. Integration by Parts

98. Change of Variable

3. Basic Classes of Integrable Functions

99. Linear Rational Functions

100. Examples

101. Ostrogradskii's Method

102. Some Irrational Functions

103. Trigonometric Functions

104. Rational Functions of x and √ax2+bx+c

105. General Remarks

Chapter VII Methods of Evaluating Definite Integrals. Improper Integrals

1. Methods of Evaluating Integrals

106. Definite Integration by Parts

107. Change of Variable in a Definite Integral

2. Approximate Methods

108. Numerical Integration

109. Graphical Integration

3. Improper Integrals

110. Integrals with Infinite Limits

111. Tests for Convergence and Divergence of Integrals with Infinite Limits

112. Integral of a Function with Infinite Jumps

113. Tests for Convergence and Divergence of Integrals of Discontinuous Functions

Chapter VIII Applications of the Integral

1. Elementary Problems and Methods of Solution

114. Method of "Summation of Elements"

115. Method of "Differential Equation". Scheme for Solution of Problems

116. Examples

2. Some Problems of Geometry and Statics. Processes of Organic Growth

117. Area of a Figure

118. Length of Arc

119. Volume of a Body

120. Area of Surface of Revolution

121. Center of Gravity and Guldin's Theorems

122. Processes of Organic Growth

Chapter IX Series

1. Numerical Series

123. Series. Convergence

124. Series with Positive Terms. Sufficient Tests for Convergence

125. Series with Arbitrary Terms. Absolute Convergence

126. Operations on Series

2. Functional Series

127. Definitions. Uniform Convergence

128. Integration and Differentiation of Functional Series

3. Power Series

129. Taylor's Series

130. Examples

131. Interval and Radius of Convergence

132. General Properties of Power Series

4. Power Series (Continued)

133. Another Method of Expanding Functions in Taylor's Series

134. Some Applications of Taylor's Series

135. Functions of a Complex Variable. Euler's Formula

Index

## Product details

- No. of pages: 508
- Language: English
- Copyright: © Pergamon 1963
- Published: January 1, 1963
- Imprint: Pergamon
- eBook ISBN: 9781483137322

## About the Author

### A. F. Bermant

## About the Editors

### I. N. Sneddon

### S. Ulam

### M. Stark

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