Functions of a real variable : elementary theory

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
To the readers......Page 6
Contents......Page 10
4. Expansions of log(1 + x), of Arc tan x and of Arc sin x      1 I\0......Page 16
1. Derivative of a vector function\0......Page 18
2. Linearity of differentiation\0......Page 20
3. Derivative of a product\0......Page 21
4. Derivative of the inverse of a function\0......Page 23
6. Derivative of an inverse function\0......Page 24
7. Derivatives of real-valued functions\0......Page 25
1. Rolle\'s Theorem\0......Page 27
2. The mean value theorem for real-valued functions\0......Page 28
3. The mean value theorem for vector functions\0......Page 30
4. Continuity of derivatives\0......Page 33
1. Derivatives of order n\0......Page 35
2. Taylor\'s formula\0......Page 36
4. CONVEX FUNCTIONS OF A REAL VARIABLE\0......Page 38
1. Definition of a convex function\0......Page 39
3. Continuity and differentiability of convex functions\0......Page 42
4. Criteria for convexity\0......Page 45
Exercises on 1\0......Page 50
Exercises on 2\0......Page 52
Exercises on 3\0......Page 54
Exercises on 4\0......Page 60
CHAPTER II PRIMITIVES AND INTEGRALS\0......Page 64
1. PRIMITIVES AND INTEGRALS\0......Page 66
1. Definition of primitives                   5 I\0......Page 0
2. Existence of primitives\0......Page 67
3. Regulated functions\0......Page 68
4. Integrals\0......Page 71
5. Properties of integrals\0......Page 74
1. Definition of an integral over a non-compact interval\0......Page 77
2. Integrals of positive functions over a non-compact interval\0......Page 81
3. Absolutely convergent integrals\0......Page 82
1  Integral of a limit of functions on a compact interval\0......Page 83
2. Integral of a limit of functions on a non-compact interval\0......Page 84
3. Normally convergent integrals\0......Page 87
4. Derivative with respect to a parameter of an integral over a compact interval\0......Page 88
5. Derivative with respect to a parameter of an integral over a non-compact interval\0......Page 90
6. Change of order of integration\0......Page 91
Exercises on 1\0......Page 94
Exercises on 2\0......Page 99
Exercises on 3\0......Page 101
CHAPTER III ELEMENTARY FUNCTIONS\0......Page 104
1. Derivatives of the exponential functions; the number e\0......Page 106
2. Derivative of log, x\0......Page 108
3. Derivatives of the circular functions; the number 7r\0......Page 109
4. Inverse circular functions\0......Page 110
5. The complex exponential\0......Page 112
6. Properties of the function e.\0......Page 113
7. The complex logarithm\0......Page 115
8. Primitives of rational functions\0......Page 116
9. Complex circular functions; hyperbolic functions\0......Page 117
1. Expansion of the real exponential\0......Page 120
2. Expansions of the complex exponential, of cos x and sin x.\0......Page 121
3. The binomial expansion\0......Page 122
Exercises on 1\0......Page 130
Exercises on 2\0......Page 140
Historical Note (Chapters 1-II-QI)\0......Page 144
Bibliography\0......Page 174
CHAPTER IV DIFFERENTIAL EQUATIONS\0......Page 176
1. The concept of a differential equation\0......Page 178
2. Differential equations admitting solutions that are primitives of regulated functions\0......Page 179
3. Existence of approximate solutions\0......Page 181
4. Comparison of approximate solutions\0......Page 183
5. Existence and uniqueness of solutions of Lipschitz and locally Lipschitz equations\0......Page 186
6. Continuity of integrals as functions of a parameter\0......Page 189
7. Dependence on initial conditions\0......Page 191
1. Existence of integrals of it linear differential equation\0......Page 192
2. Linearity of the integrals of a linear differential equation\0......Page 194
3. Integrating the mhomogeneous linear equation\0......Page 197
4. Fundamental systems of integrals of a linear system of scalar differential equations\0......Page 198
5. Adjoint equation\0......Page 201
6. Linear differential equations with constant coefficients\0......Page 203
7. Linear equations of order ii\0......Page 207
8 Linear equations of order n with constant coefficients\0......Page 209
9 Systems of linear equations with constant coefficients\0......Page 211
Exercises on 1\0......Page 214
Exercises on 2\0......Page 219
Historical Note\0......Page 222
CHAPTER V LOCAL STUDY OF FUNCTIONS\0......Page 224
1. Comparison relations: 1. Weak relations\0......Page 226
2. Comparison relations: II. Strong relations\0......Page 229
4. Comparison relations between strictly positive functions\0......Page 232
5. Notation\0......Page 234
1. Scales of comparison\0......Page 235
2. Principal parts and asymptotic expansions\0......Page 236
3. Sums and products of asymptotic expansions\0......Page 238
4. Composition of asymptotic expansions\0......Page 239
5. Asymptotic expansions with variable coefficients\0......Page 241
3. ASYMPTOTIC EXPANSIONS OF FUNCTIONS OF A REAL VARIABLE\0......Page 242
1. Integration of comparison relations: I. Weak relations\0......Page 243
2. Application: logarithmic criteria for convergence of integrals\0......Page 244
3. Integration of comparison relations:  Strong relations\0......Page 245
4. Differentiation of comparison relations\0......Page 247
5. Principal part of a primitive\0......Page 248
6. Asymptotic expansion of a primitive\0......Page 250
1. Convergence criteria for series with positive terms\0......Page 251
2. Asymptotic expansion of the partial sums of a series\0......Page 253
3. Asymptotic expansion of the partial products of an infinite product\0......Page 258
4. Application: convergence criteria of the second kind for series with positive terms\0......Page 259
1. Hardy fields\0......Page 262
2. Extension of a Hardy field\0......Page 263
3. Comparison of functions in a Hardy field\0......Page 265
4. (H) Functions\0......Page 267
5. Exponentials and iterated logarithms\0......Page 268
6. Inverse function of an (H) function\0......Page 270
Exercises on 1\0......Page 274
Exercises on 2\0......Page 275
Exercises on 3\0......Page 276
Exercises on Appendix\0......Page 278
CHAPTER VI GENERALIZED TAYLOR EXPANSIONS. EULER-MACLAURIN SUMMATION FORMULA\0......Page 282
1  Composition operators on an algebra of polynomials\0......Page 284
2. Appell polynomials attached to a composition operator\0......Page 287
3. Generating series for the Appell polynomials\0......Page 289
4. Bernoulli polynomials\0......Page 290
5. Composition operators on functions of a real variable\0......Page 292
6. Indicatrix of a composition operator\0......Page 293
7. The Euler-Maclaurin summation formula\0......Page 297
1. Eulerian expansion of cot z\0......Page 298
2. Eulerian expansion of sin\0......Page 301
3. Application to the Bernoulli numbers\0......Page 302
1. Bounds for the remainder in the Euler-Maclaurin summation formula\0......Page 303
2. Application to asymptotic expansions\0......Page 304
Exercises on 1\0......Page 306
Exercises on 2\0......Page 307
Exercises on 3\0......Page 311
Historical Note (Chapters V and VI)\0......Page 314
CHAPTER VII THE GAMMA FUNCTION\0......Page 318
1. Definition of the Gamma function\0......Page 320
2. Properties of the Gamma function\0......Page 322
3. The Euler integrals\0......Page 325
1. Extending the Gamma function to C\0......Page 330
2. The complements\' relation and the Legendre-Gauss multiplication formula\0......Page 331
3. Stirling\'s expansion\0......Page 334
Exercises on 1\0......Page 340
Exercises on 1\0......Page 342
Historical Note\0......Page 344
Bibliography\0......Page 346
INDEX OF NOTATION\0......Page 348
INDEX\0......Page 350