Advanced Calculus

CHAPTER I REAL NUMBERS. SECTIONS

1. The Continuous Variable
2. Continuity of a Straight Line
3. The Rational Number
4. Real Numbers
5. Properties of the Real Number
6. Laws of Operation. Absolute Value of a Number
7. Sections of the Real Numbers
8. Decimal Representation
9. Correspondence of Number and Point. Continuous Variable. Interval
1O. Roots. Indices
11. Inequalities
EXERCISES I. Lagrange’s Identity. Schwarz’s Inequality

CHAPTER II SETS. SEQUENCES. LIMITING POINTS. LIMITS

12. Sets. Sequences
13. The Upper and Lower Bounds
14. Limits. Notation. Null Sequence
15. Monotonic Functions
16. Sequence of Intervals. Method of the Decreasing Interval
17. Limiting Points or Points of Condensation. The Bolzano Weierstrass Theorem
18. Limits of Indetermination
19. Existence of a Limit. Sequence
20. Examples. Cauchy’s First and Second Theorems
21. Existence of a Limit. Function of 1:. Continuity of Function
22. Exponential Functions
23. Logarithms
24. The Base e. Theorems in Limits. Derivative of x^n when n is Irrational
25. Limits and Inequalities
26. Extension of Range of Definition
EXERCISES II. Euler’s Constant. Stirling’s Approximation

CHAPTER III FUNCTIONS OF ONE VARIABLE. DERIVATIVES. DIFFERENTIALS

27. Oscillation of a Function
28. Theorems on Continuous Functions. Uniform Continuity
29. Discontinuity. Removable Discontinuities. Discontinuities of the First and Second Kinds
30. Derivatives
31. Elementary Functions. Function of a Function
32. Inverse Functions
33. Rolle’s Theorem
34. Theorem of Mean Value. Lipschitz’s Condition
EXERCISES III
35. Difierentials
36. Higher Derivatives
37. Other Methods for Higher Derivatives. Rodrigues’ Formula
EXERCISES IV
38. Derivative of a Determinant
39. Linear Dependence of Functions. Wronskians

CHAPTER IV FUNCTIONS OF SEVERAL VARIABLES. DERIVATIVES. DIFFERENTIALS. CHANGE OF VARIABLES

40. Functions of more than one Variable. Limiting Points
41. Limits and Continuity
42. Sequence of Decreasing Regions
43. Theorems on Continuous Functions
44. Function of Functions
45. Partial Derivatives. Mean Value Theorem
46. Differentials
47. Higher Differentials
48. Change of Variables
49. Special Cases
50. Elimination of Functions
EXERCISES V

CHAPTER V IMPLICIT FUNCTIONS. JACOBIANS

51. Implicit Functions
52. Existence Theorem I. Analytical Continuation
53. Derivatives of Implicit Function
54. Existence Theorem II
55. The Jacobian
56. Existence Theorem III. Inversion
57. Dependence of Functions
58. The Hessian
EXERCISES VI

CHAPTER VI INFINITE SERIES. COMPLEX FUNCTIONS OF A REAL VARIABLE

59. Infinite Series. Derangement of Terms. Dirichlet’s Theorem. Conditional Convergence
60. Tests of Convergence. Kummer’s Test. Raabe’s Teet. Gauss’s Test.
Cauchy’s Condensation Test. The Hypergeometric Series. Cauchy’s Test
61. Tests of Abel and Dirichlet
62. Uniform Convergence. The M -Test. Abel’s Test. Dirichlet’s Test
63. Tannery’s Theorem
64. Abel’s Theorem
65. Cesaro’s Theorem
66. Derangement of a Series. Multiplication of Series. Legendre’s Coefficients
EXERCISES VII
67. Series of Complex Terms. Absolute Convergence. Tannery’s Theorem. Derangement of Series
68. The Exponential Function
69. Trigonometric and Hyperbolic Functions. Euler’s Expressions for the Sine and Cosine. Periodicity of e^z
70. Logarithms
71. Inverse Trigonometric Functions
72. The Generalised Power
73. Complex Functions of a Real Variable
74. Logarithmic and Binomial Series
75. Uniform Convergence
EXERCISES VIII

CHAPTER VIISUBSTITUTION OF A SERIES IN A SERIES. REVERSION SERIES.
LAGRANGE’S EXPANSION. MAXIMA A MINIMA OF FUNCTIONS OF SEVERAL VARIABLES

76. Power Series. Substitution of a Power Series in a Power Series
77. Division by a Series
78. Reversion of Series
79. Lagrange’s Expansion
80. Examples. Differentiation of Lagrange’s Expansion. Rodrigues’ Formula for P”(x)
81. Implicit Function of one Variable. Factorisation
82. Algebraic Forms
83. Remainder in Taylor’s Theorem
84. Maxima and Minima
85. Absolute Maxima and Minima. Turning Values of an Implicit Function
86. Hadamard’s Determinant
EXERCISES IX

CHAPTER VIII INFINITE PRODUCTS. PRODUCTS AND SERIES OF PARTIAL
FRACTIONS FOR TRIGONOMETRIC FUNCTIONS. GAMMA FUNCTIONS

87. Infinite Products
88. Tests for Convergence of Products. Absolute Convergence
89. Derangement of Factors in Product
EXERCISES X
90. Uniform Convergence. Differentiation
91. Tannery’s Theorem
92. Infinite Products for Trigonometric and Hyperbolic Functions
93. Expansions in Partial Fractions
94. Bernoulli’s Numbers
EXERCISES XI. Euler’s Numbers
95. The Gamma. Function. Gauss’s II-Function
96. Properties of Г(x)
97. Gauss’s Function Ψ(x)
98. Examples
99. The Hypergeometric Function
100. Gauss’s Formula for Г(mm)
EXERCISES XII

CHAPTER IX INTEGRATION OF BOUNDED FUNCTIONS

101. Intervals. Sets. Consecutive Divisions
102. The Sums S and s
103. Darboux’s Theorem
104. Functions with Limited Variation
105. The Definite Integral
106. Condition of Integrability
107. Other Forms of the Definition of an Integral
108. Integrable Functions. Continuous Functions. Monotonic Functions.
Functions with Limited Variation
109. General Theorems
110. Discontinuities
111. Properties of the Integral. First and Second Theorems of Mean Value
112. The Integral as a Function of its Limits. The Indefinite
Integral. Discontinuity of Integrand
113. Examples
114. Transformations of the Integral
EXERCISES XTII. The Legendre Polynomials. Schwarz’s Inequality

CHAPTER X RECTIFICATION. CURVILINEAR INTEGRALS. AREAS.
REPEATED AND DOUBLE INTEGRALS. VOLUMES. SURFACES

115. Rectification of Curves
116. Curvilinear Integrals
117. Area
118. Area of a Closed Curve. Conditions to be satisfied by a Curve
EXERCISES XIV
119. Integral as Function of a Parameter. Discontinuities of Integrand
120. Continuity with respect to a Parameter
121. Differentiation and Integration
122. Double Integrals. Darboux’s Theorem
123. Division of the Area
124. Integrable Functions
125. General Theorems
126. Reduction to Repeated Integrals
126A. Another Proof
127. Conditions for Repeated Integrals
128. Volume. Area of a Curved Surface
129. Curves on a Surface. Element of Surface. Change of Axes.
Change of Parameters
130. Worked Examples
131. Green’s Theorem
EXERCISES XV

CHAPTER XI MULTIPLE INTEGRALS. SURFACE INTEGRALS

132. Multiple Integrals. Integrable Functions
133. Change of Variables
134. General Method
135. Surface Integrals
136. Surfaces Positive and Negative Sides. Convention as to Sign
137. Stokes’s Theorem
138. Green’s General Theorem
139. Transformation of V217
140. Worked Examples
EXERCISES XVI. Elliptic Coordinates

CHAPTER XII IMPROPER INTEGRALS

141. Improper Integrals. Singular Point
142. Definitions. Range Finite. Principal Value. Range Infinite. Absolute Convergence
143. General Conditions for Convergence. Change of Values of the Integrand. Singular Integral
144. Special Test. Convergence at a Singular Point. Convergence at Infinity
145. General Theorems. Integration of Products. Absolute Convergence.
Integral as Function of its Limits. Abel’s Theorem. Dirichlet’s Theorem.
First and Second Theorems of Mean Value. Transformations of the Improper Integral. Beta Function
146. Worked Examples. Additional Tests. The Comparison Test. Frullani’s Integral
147. Complex Functions of a Real Variable. Definition of Gamma Function as an Integral
148. Miscellaneous Examples. Integral as Limit of a Sum. Integral Test for Convergence of Series
EXERCISES XVII. Ermakoff’s Tests

CHAPTER XIII IMPROPER INTEGRALS: REPEATED AND DOUBLE INTEGRALS, FIELD OF INTEGRATION FINITE

149. Improper Double Integrals. Points and Curves of Infinite Discontinuity
150. Absolute Convergence
151. Uniform Convergence. Discontinuities. Uniform Convergence in General
152. Continuity of Integrals
153. Change of Order of Integration
154. Change of Order : Variable Limits
155. Differentiation under the Integral Sign
156. Evaluation of Improper Double Integrals
157. Multiple Integrals. Change of Variables
EXERCISES XVIII

CHAPTER XIV DOUBLE INTEGRALS: RANGE INFINITE

158. Range of Integration Infinite. p Uniform Convergence.
The M -Test. Abel’s Test. Dirichlet’s Test .
159. Continuity of Integrals. Limits of Integrals. Analogue
of Tannery’s Theorem .
160. Repeated Integrals : One Limit Infinite
161. Differentiation under the Integral Sign
EXERCISES XIX
162. Repeated Integrals : Infinite Limits. Beta Function
163. Double Integrals With Infinite Limits
EXERCISES XX

CHAPTER XV INTEGRATION OF SERIES. GAMMA FUNCTIONS

164. Integration of Series
165. General Theorems. Integrals for Bernoulli’s Numbers
EXERCISES XXI
166. Integrals for Euler’s Constant
167. Integral for log Г (x). Binet’s Function μ(x)
168. Asymptotic Expansion for log Г (x). Integral for μ(x)
169. Gauss’s Function Ψ(x)
170. Another Proof of the Integral for log Г(x)
171. Minimum Value of Г(x)
172. Integrals reducible to Gamma Functions. Dirichlet’s and
Liouville’s Integral A
EXERCISES XXII
EXERCISES XXIII

INDEX