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# Algebraical numbers 41 chapter ii functions of real

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Algebraical numbers, 41. CHAPTER II FUNCTIONS OF REAL VARIABLES 20. The idea of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 21. The graphical representation of functions. Coordinates . . . . . . . . 46 22. Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 23. Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 24–25. Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 26–27. Algebraical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 28–29. Transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 30. Graphical solution of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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CONTENTS viii SECT. PAGE 31. Functions of two variables and their graphical representation . . 68 32. Curves in a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 33. Loci in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Trigonometrical functions, 60. Arithmetical functions, 63. Cylinders, 72. Contour maps, 72. Cones, 73. Surfaces of revolution, 73. Ruled sur- faces, 74. Geometrical constructions for irrational numbers, 77. Quadra- ture of the circle, 79. CHAPTER III COMPLEX NUMBERS 34–38. Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 39–42. Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 43. The quadratic equation with real coefficients . . . . . . . . . . . . . . . . . . 96 44. Argand’s diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 45. De Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 46. Rational functions of a complex variable . . . . . . . . . . . . . . . . . . . . . . 104 47–49. Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Properties of a triangle, 106, 121. Equations with complex coeffi- cients, 107. Coaxal circles, 110. Bilinear and other transforma- tions, 111, 116, 125. Cross ratios, 114. Condition that four points should be concyclic, 116. Complex functions of a real variable, 116. Construction of regular polygons by Euclidean methods, 120. Imaginary points and lines, 124. CHAPTER IV LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE 50. Functions of a positive integral variable . . . . . . . . . . . . . . . . . . . . . . . 128 51. Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 52. Finite and infinite classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
CONTENTS ix SECT. PAGE 53–57. Properties possessed by a function of n for large values of n . . . 131 58–61. Definition of a limit and other definitions . . . . . . . . . . . . . . . . . . . . . 138 62. Oscillating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 63–68. General theorems concerning limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 69–70. Steadily increasing or decreasing functions . . . . . . . . . . . . . . . . . . . . 157 71. Alternative proof of Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . . 159 72. The limit of x n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 73. The limit of 1 + 1 n n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 74. Some algebraical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 75. The limit of n ( n x - 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 76–77. Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 78. The infinite geometrical series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 79. The representation of functions of a continuous real variable by means of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 80. The bounds of a bounded aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 81. The bounds of a bounded function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 82. The limits of indetermination of a bounded function . . . . . . . . . . 180 83–84. The general principle of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 183 85–86. Limits of complex functions and series of complex terms . . . . . . 185 87–88. Applications to z n and the geometrical series . . . . . . . . . . . . . . . . . 188 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Oscillation of sin nθπ , 144, 146, 181. Limits of n k x n , n x , n n , n n !, x n n ! , m n x n , 162, 166. Decimals, 171. Arithmetical series, 175. Harmonical series, 176. Equation x n +1 = f ( x n ), 190. Expansions of rational func- tions, 191. Limit of a mean value, 193. CHAPTER V LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS 89–92. Limits as x → ∞ or x → -∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

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CONTENTS x SECT. PAGE 93–97. Limits as x a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 98–99. Continuous functions of a real variable . . . . . . . . . . . . . . . . . . . . . . . . 210 100–104. Properties of continuous functions. Bounded functions. The oscillation of a function in an interval . . . . . . . . . . . . . . . . . . . . 215 105–106. Sets of intervals on a line. The Heine-Borel Theorem . . . . . . . . . . 223 107. Continuous functions of several variables . . . . . . . . . . . . . . . . . . . . . . 228 108–109.
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