XX
Contents
16 - Comparison relations. Criteria of Cauchy and d'Alembert 132
17 - Infinite limits ...................................... 138
18 - Unconditional convergence: associativity .............. 139
§3. First concepts of analytic functions ......................... 148
19 - The Taylor series ................................... 148
20 - The principle of analytic continuation ................. 158
21 - The function cot x and the series L l/n 2k . . . . . . . . . . . . . 162
22 - Multiplication of series. Composition of analytic functions. Formal series ............................... 167
23 - The elliptic functions of Weierstrass ..... . . . . . . . . . . . .. 178
III - Convergence: Continuous variables ...................... 187
§1. The intermediate value theorem ........................... 187
1 - Limit values of a function. Open and closed sets ........ 187
2 - Continuous functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 192
3 - Right and left limits of a monotone function ............ 197
4 - The intermediate value theorem ....................... 200
§2. Uniform convergence ..................................... 205
5 - Limits of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . 205
6 - A slip up of Cauchy's ................................ 211
7 - The uniform metric ................................. 216
8 - Series of continuous functions. Normal convergence ...... 220
§3. Bolzano-Weierstrass and Cauchy's criterion ................. 225
9 - Nested intervals, Bolzano-Weierstrass, compact sets ..... 225
10 - Cauchy's general convergence criterion ................ 228
11 - Cauchy's criterion for series: examples ................ 234
12 - Limits of limits .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
13 - Passing to the limit in a series of functions ............ 241
§4. Differentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
14 - Derivatives of a function ............................ 244
15 - Rules for calculating derivatives ...................... 252
16 - The mean value theorem ............................ 260
17 - Sequences and series of differentiable functions. . . . . . . . . 265
18 - Extensions to unconditional convergence .............. 270
§5. Differentiable functions of several variables .................. 273
19 - Partial derivatives and differentials ................... 273
20 - Differentiability of functions of class C 1 . . . . . . . . . . . . . . . 276
21 - Differentiation of composite functions ................. 279
22 - Limits of differentiable functions ..................... 284
23 - Interchanging the order of differentiation . . . . . . . . . . . . . . 287
24 - Implicit functions .................................. 290
Contents
16 - Comparison relations. Criteria of Cauchy and d'Alembert 132
17 - Infinite limits ...................................... 138
18 - Unconditional convergence: associativity .............. 139
§3. First concepts of analytic functions ......................... 148
19 - The Taylor series ................................... 148
20 - The principle of analytic continuation ................. 158
21 - The function cot x and the series L l/n 2k . . . . . . . . . . . . . 162
22 - Multiplication of series. Composition of analytic functions. Formal series ............................... 167
23 - The elliptic functions of Weierstrass ..... . . . . . . . . . . . .. 178
III - Convergence: Continuous variables ...................... 187
§1. The intermediate value theorem ........................... 187
1 - Limit values of a function. Open and closed sets ........ 187
2 - Continuous functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 192
3 - Right and left limits of a monotone function ............ 197
4 - The intermediate value theorem ....................... 200
§2. Uniform convergence ..................................... 205
5 - Limits of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . 205
6 - A slip up of Cauchy's ................................ 211
7 - The uniform metric ................................. 216
8 - Series of continuous functions. Normal convergence ...... 220
§3. Bolzano-Weierstrass and Cauchy's criterion ................. 225
9 - Nested intervals, Bolzano-Weierstrass, compact sets ..... 225
10 - Cauchy's general convergence criterion ................ 228
11 - Cauchy's criterion for series: examples ................ 234
12 - Limits of limits .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
13 - Passing to the limit in a series of functions ............ 241
§4. Differentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
14 - Derivatives of a function ............................ 244
15 - Rules for calculating derivatives ...................... 252
16 - The mean value theorem ............................ 260
17 - Sequences and series of differentiable functions. . . . . . . . . 265
18 - Extensions to unconditional convergence .............. 270
§5. Differentiable functions of several variables .................. 273
19 - Partial derivatives and differentials ................... 273
20 - Differentiability of functions of class C 1 . . . . . . . . . . . . . . . 276
21 - Differentiation of composite functions ................. 279
22 - Limits of differentiable functions ..................... 284
23 - Interchanging the order of differentiation . . . . . . . . . . . . . . 287
24 - Implicit functions .................................. 290