An Analytical Calculus: Volume 2: For School and University

Cover......Page 1
Title......Page 4
Copyright......Page 5
CONTENTS......Page 6
PREFACE......Page 8
1. The logarithm......Page 10
2. First properties of the logarithm......Page 11
3. The graph of log x......Page 13
4. The use of logarithms in differentiation......Page 19
5. The use of logarithms in integrating simple rational functions......Page 20
6. The exponential function......Page 27
7. The relations dy/dx = 1/x, dy/dx =y......Page 29
8. The integration of ex......Page 33
9. The reconciliation of loge x and log10 x......Page 36
1. A series giving sin x......Page 47
2. A series giving 1/(1 + x)......Page 49
3. Expansion in series......Page 50
4. The coefficients in an infinite series......Page 52
5. Taylor\'s theorem......Page 53
7. Maclaurin\'s series......Page 58
8. The series for sin x and cos x......Page 59
9. The binomial series......Page 60
10. The logarithmic series......Page 62
11. The exponential series......Page 63
12. Approximations......Page 64
13. Newton\'s approximation to a root of an equation......Page 65
14. Leibniz\'s theorem......Page 71
1. The hyperbolic cosine and sine......Page 93
2. Other hyperbolic functions......Page 96
3. The graph y = cosh x......Page 103
4. The graph y = sinh x......Page 104
5. The inverse hyperbolic cosine......Page 105
6. The inverse hyperbolic sine......Page 107
1. Parametric representation......Page 109
2. The sense of description of a curve......Page 110
3. The \'length\' postulate......Page 111
4. The length of a curve......Page 112
5. The length of a curve in Cartesian coordinates......Page 113
6. The length of a curve in polar coordinates......Page 114
7. The \'gradient angle\' Ψ......Page 116
8. The angle from the radius vector to the tangent......Page 118
9. The perpendicular on the tangent......Page 119
10. Other coordinate systems......Page 120
11. Curvature......Page 122
12. A parametric form for a curve in terms of s......Page 125
13. Newton\'s formula......Page 126
14. The circle of curvature......Page 128
15. Envelopes......Page 132
17. The area of a closed curve......Page 137
18.* Second theorem of Pappus......Page 141
1. Introduction......Page 167
2. Definitions......Page 170
3. Addition, subtraction, multiplication......Page 172
4. Division......Page 173
5. Equal complex numbers......Page 174
6. The complex number as a number-pair......Page 175
7. The Argand diagram......Page 177
8. Modulus and argument......Page 178
9. The representation in an Argand diagram of the sum of two numbers......Page 180
11. The product of two complex numbers......Page 181
12. The product in an Argand diagram......Page 182
13. De Moivre\'s theorem......Page 183
15. Complex powers......Page 187
16. Pure imaginary powers......Page 188
17. Multiplicity of values......Page 190
18, The logarithm of a complex number, to the base e......Page 192
19. The sine and cosine......Page 193
21. The differentiation and integration of complex numbers......Page 195
2. Rational functions......Page 209
4. Integrals involving √(ax2 + 2hx + b)......Page 212
5. Integrals involving √(ax2 + 2hx + b): alternative treatment......Page 216
6. Trigonometric functions......Page 223
8. Exponential times polynomial in sines and cosines of multiples of x......Page 224
1. \'Infinite\' limits of integration......Page 226
2. \'Infinite\' integrand......Page 231
APPENDIX......Page 245
CHAPTER VII......Page 254
CHAPTER VIII......Page 260
CHAPTER IX......Page 265
CHAPTER X......Page 266
CHAPTER XI......Page 271
CHAPTER XIII......Page 276
INDEX......Page 280