Absolutely convergent series 418 186187 conditionally

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184–185. Absolutely convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 186–187. Conditionally convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 188. Alternating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 189. Abel’s and Dirichlet’s tests of convergence . . . . . . . . . . . . . . . . . . . . 425 190. Series of complex terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 191–194. Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 195. Multiplication of series in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 The series n k r n and allied series, 385. Transformation of infinite inte- grals by substitution and integration by parts, 404, 406, 413. The series a n cos , a n sin , 419, 425, 427. Alteration of the sum of a series by rearrangement, 423. Logarithmic series, 431. Binomial series, 431, 433. Multiplication of conditionally convergent series, 434, 439. Recurring se- ries, 437. Difference equations, 438. Definite integrals, 441. Schwarz’s inequality for infinite integrals, 442.
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CONTENTS xiii CHAPTER IX THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A REAL VARIABLE SECT. PAGE 196–197. The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 198. The functional equation satisfied by log x . . . . . . . . . . . . . . . . . . . . . 447 199–201. The behaviour of log x as x tends to infinity or to zero . . . . . . . . 448 202. The logarithmic scale of infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 203. The number e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 204–206. The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 207. The general power a x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 208. The exponential limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 209. The logarithmic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 210. Common logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 211. Logarithmic tests of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 212. The exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 213. The logarithmic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 214. The series for arc tan x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 215. The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 216. Alternative development of the theory . . . . . . . . . . . . . . . . . . . . . . . . 482 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Integrals containing the exponential function, 460. The hyperbolic func- tions, 463. Integrals of certain algebraical functions, 464. Euler’s con- stant, 469, 486. Irrationality of e , 473. Approximation to surds by the bi- nomial theorem, 480. Irrationality of log 10 n , 483. Definite integrals, 491. CHAPTER X THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS 217–218. Functions of a complex variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 219. Curvilinear integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 220. Definition of the logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . 497 221. The values of the logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . 499
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CONTENTS xiv SECT. PAGE 222–224. The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 225–226. The general power a z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 227–230. The trigonometrical and hyperbolic functions . . . . . . . . . . . . . . . . . 512 231. The connection between the logarithmic and inverse trigonomet- rical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 232. The exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 233. The series for cos z and sin z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 234–235. The logarithmic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 236. The exponential limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 237. The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 The functional equation satisfied by Log z , 503. The function e z , 509. Logarithms to any base, 510. The inverse cosine, sine, and tangent of a complex number, 516. Trigonometrical series, 523, 527, 540. Roots of transcendental equations, 534. Transformations, 535, 538. Stereographic projection, 537. Mercator’s projection, 538. Level curves, 539. Definite integrals, 543. Appendix I. The proof that every equation has a root . . . . . . . . . . . . . . . 545 Appendix II. A note on double limit problems . . . . . . . . . . . . . . . . . . . . . . . . 553 Appendix III. The circular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Appendix IV. The infinite in analysis and geometry . . . . . . . . . . . . . . . . . . . 560
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CHAPTER I REAL VARIABLES 1. Rational numbers. A fraction r = p/q , where p and q are pos- itive or negative integers, is called a rational number . We can suppose (i) that p and q have no common factor, as if they have a common factor
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