Contents
XXI
Appendix to Chapter III ..................................... 303
1 - Cartesian spaces and general metric spaces . . . . . . . . . . . . . 303
2 - Open and closed sets ................... . . . . . . . . . . . . . 306
3 - Limits and Cauchy's criterion in a metric space; complete
spaces .......................................... 308
4 - Continuous functions ................................ 311
5 - Absolutely convergent series in a Banach space ......... 313
6 - Continuous linear maps .............................. 316
7 - Compact spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
8 - Topological spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
IV
Powers, Exponentials, Logarithms, Trigonometric Functions ....................................... " ............. 325
§ 1. Direct construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
1 - Rational exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
2 - Definition of real powers .......... . . . . . . . . . . . . . . . . . . . 327
3 - The calculus of real exponents ........................ 330
4 - Logarithms to base a. Power functions . . . . . . . . . . . . . . . . . 332
5 - Asymptotic behaviour .................... . . . . . . . . . . . 333
6 - Characterisations of the exponential, power and logarithmic functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
7 - Derivatives of the exponential functions: direct method .. 339
8 - Derivatives of exponential functions, powers and logarithms342
§2. Series expansions ........................................ 345
9 - The number e. Napierian logarithms ................... 345
10 - Exponential and logarithmic series: direct method ...... 346
11 - Newton's binomial series ............................ 351
12 - The power series for the logarithm ................... 359
13 - The exponential function as a limit ................... 368
14 - Imaginary exponentials and trigonometric functions .... 372
15 - Euler's relation chez Euler ........................... 383
16 - Hyperbolic functions ............................... 388
§3. Infinite products ......................................... 394
17 - Absolutely convergent infinite products ............... 394
18 - The infinite product for the sine function .............. 397
19 - Expansion of an infinite product in series .............. 403
20 - Strange identities .................................. 407
§4. The topology of the functions Arg(z) and .cog z ............. 414
Index ......................................................... 425
XXI
Appendix to Chapter III ..................................... 303
1 - Cartesian spaces and general metric spaces . . . . . . . . . . . . . 303
2 - Open and closed sets ................... . . . . . . . . . . . . . 306
3 - Limits and Cauchy's criterion in a metric space; complete
spaces .......................................... 308
4 - Continuous functions ................................ 311
5 - Absolutely convergent series in a Banach space ......... 313
6 - Continuous linear maps .............................. 316
7 - Compact spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
8 - Topological spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
IV
Powers, Exponentials, Logarithms, Trigonometric Functions ....................................... " ............. 325
§ 1. Direct construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
1 - Rational exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
2 - Definition of real powers .......... . . . . . . . . . . . . . . . . . . . 327
3 - The calculus of real exponents ........................ 330
4 - Logarithms to base a. Power functions . . . . . . . . . . . . . . . . . 332
5 - Asymptotic behaviour .................... . . . . . . . . . . . 333
6 - Characterisations of the exponential, power and logarithmic functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
7 - Derivatives of the exponential functions: direct method .. 339
8 - Derivatives of exponential functions, powers and logarithms342
§2. Series expansions ........................................ 345
9 - The number e. Napierian logarithms ................... 345
10 - Exponential and logarithmic series: direct method ...... 346
11 - Newton's binomial series ............................ 351
12 - The power series for the logarithm ................... 359
13 - The exponential function as a limit ................... 368
14 - Imaginary exponentials and trigonometric functions .... 372
15 - Euler's relation chez Euler ........................... 383
16 - Hyperbolic functions ............................... 388
§3. Infinite products ......................................... 394
17 - Absolutely convergent infinite products ............... 394
18 - The infinite product for the sine function .............. 397
19 - Expansion of an infinite product in series .............. 403
20 - Strange identities .................................. 407
§4. The topology of the functions Arg(z) and .cog z ............. 414
Index ......................................................... 425