The Fundamentals of Mathematical Analysis

Chapter 15 Series of Numbers §1. Introduction 234. Elementary Concepts 235. The Most Elementary Theorems §2. The Convergence of Positive Series 236. A Condition for the Convergence of a Positive Series 237. Theorems on the Comparison of Series 238. Examples 239. Cauchy's and d'Alembert's Tests 240. Raabe's Test 241. The Maclaurin-Cauchy Integral Test §3. The Convergence of Arbitrary Series 242. The Principle of Convergence 243. Absolute Convergence 244. Alternating Series §4. The Properties of Convergent Series 245. The Associative Property 246. The Permuting Property of Absolutely Convergent Series 247. The Case of Non-Absolutely Convergent Series 248. The Multiplication of Series §5. Infinite Products 249. Fundamental Concepts 250. The Simplest Theorems. The Connection with Series 251. Examples §6. The Expansion of Elementary Functions in Power Series 252. Taylor Series 253. The Expansion of the Exponential and Elementary Trigonometrical Functions in Power Series 254. Euler's Formulae 255. The Expansion for the Inverse Tangent 256. Logarithmic Series 257. Stirling's Formula 258. Binomial Series 259. A Remark on the Study of the Remainder §7. Approximate Calculations Using Series 260. Statement of the Problem 261. The Calculation of the Number π 262. The Calculation of LogarithmsChapter 16 Sequences and Series of Functions §1. Uniform Convergence 263. Introductory Remarks 264. Uniform and Non-Uniform Convergence 265. The Condition for Uniform Convergence §2. The Functional Properties of the Sum of a Series 266. The Continuity of the Sum of a Series 267. The Case of Positive Series 268. Termwise Transition to a Limit 269. Termwise Integration of Series 270. Termwise Differentiation of Series 271. An Example of a Continuous Function Without a Derivative §3. Power Series and Series of Polynomials 272. The Interval of Convergence of a Power Series 273. The Continuity of the Sum of a Power Series 274. Continuity at the End Points of the Interval of Convergence 275. Termwise Integration of a Power Series 276. Termwise Differentiation of a Power Series 277. Power Series as Taylor Series 278. The Expansion of a Continuous Function in a Series of Polynomials §4. An Outline of the History of Series 279. The Epoch of Newton and Leibniz 280. The Period of the Formal Development of the Theory of Series 281. The Creation of a Precise TheoryChapter 17 Improper Integrals §1. Improper Integrals with Infinite Limits 282. The Definition of Integrals with Infinite Limits 283. The Application of the Fundamental Formula of Integral Calculus 284. An Analogy with Series. Some Simple Theorems 285. The Convergence of the Integral in the Case of a Positive Function 286. The Convergence of the Integral in the General Case 287. More Refined Tests §2. Improper Integrals of Unbounded Functions 288. The Definition of Integrals of Unbounded Functions 289. An Application of the Fundamental Formula of Integral Calculus 290. Conditions and Tests for the Convergence of an Integral §3. Transformation and Evaluation of Improper Integrals 291. Integration by Parts in the Case of Improper Integrals 292. Change of Variables in Improper Integrals 293. The Evaluation of Integrals by Artificial MethodsChapter 18 Integrals Depending on a Parameter §1. Elementary Theory 294. Statement of the Problem 295. Uniform Approach to a Limit Function 296. Taking Limits Under the Integral Sign 297. Differentiation Under the Integral Sign 298. Integration Under the Integral Sign 299. The Case When the Limits of the Integral Also Depend on the Parameter 300. Examples §2. Uniform Convergence of Integrals 301. The Definition of Uniform Convergence of Integrals 302. Conditions and Sufficiency Tests for Uniform Convergence 303. The Case of Integrals with Finite Limits §3. The Use of the Uniform Convergence of Integrals 304. Taking Limits Under the Integral Sign 305. The Integration of an Integral with Respect to the Parameter 306. The Differentiation of an Integral with Respect to the Parameter 307. A Remark on Integrals with Finite Limits 308. The Evaluation of Some Improper Integrals §4. Eulerian Integrals 309. The Eulerian Integral of the First Type 310. The Eulerian Integral of the Second Type 311. Some Simple Properties of the Γ Function 312. Examples 313. Some Historical Remarks on Changing the Order of Two Limit OperationsChapter 19 Implicit Functions. Functional Determinants §1. Implicit Functions 314. The Concept of an Implicit Function of One Variable 315. The Existence and Properties of an Implicit Function 316. An Implicit Function of Several Variables 317. The Determination of Implicit Functions from a System of Equations 318. The Evaluation of Derivatives of Implicit Functions §2. Some Applications of the Theory of Implicit Functions 319. Relative Extremes 320. Lagrange's Method of Undetermined Multipliers 321. Examples and Problems 322. The Concept of the Independence of Functions 323. The Rank of a Functional Matrix §3. Functional Determinants and their Formal Properties 324. Functional Determinants 325. The Multiplication of Functional Determinants 326. The Multiplication of Non-Square Functional MatricesChapter 20 Curvilinear Integrals §1. Curvilinear Integrals of the First Kind 327. The Definition of a Curvilinear Integral of the First Kind 328. The Reduction to an Ordinary Definite Integral 329. Examples §2. Curvilinear Integrals of the Second Kind 330. The Definition of Curvilinear Integrals of the Second Kind 331. The Existence and Evaluation of a Curvilinear Integral of the Second Kind 332. The Case of a Closed Contour. The Orientation of the Plane 333. Examples 334. The Connection Between Curvilinear Integrals of Both Kinds 335. Applications to Physical ProblemsChapter 21 Double Integrals §1. The Definition and Simplest Properties of Double Integrals 336. The Problem of the Volume of a Cylindrical Body 337. The Reduction of a Double Integral to a Repeated Integral 338. The Definition of a Double Integral 339. A Condition for the Existence of a Double Integral 340. Classes of Integrable Functions 341. The Properties of Integrable Functions and Double Integrals 342. An Integral as an Additive Function of the Domain; Differentiation in the Domain §2. The Evaluation of a Double Integral 343. The Reduction of a Double Integral to a Repeated Integral in the Case of a Rectangular Domain 344. The Reduction of a Double Integral to a Repeated Integral in the Case of a Curvilinear Domain 345. A Mechanical Application §3. Green's Formula 346. The Derivation of Green's Formula 347. An Expression for Area by Means of Curvilinear Integrals §4. Conditions for a Curvilinear Integral to be Independent of the Path of Integration 348. The Integral Along a Simple Closed Contour 349. The Integral Along a Curve Joining Two Arbitrary Points 350. The Connection with the Problem of Exact Differentials 351. Applications to Physical Problems §5. Change of Variables in Double Integrals 352. Transformation of Plane Domains 353. An Expression for Area in Curvilinear Coordinates 354. Additional Remark 355. A Geometrical Derivation 356. Change of Variables in Double Integrals 357. The Analogy with a Simple Integral. The Integral Over an Oriented Domain 358. Examples 359. Historical NoteChapter 22 The Area of a Surface. Surface Integrals §1. Two-Sided Surfaces 360. Parametric Representation of a Surface 361. The Side of a Surface 362. The Orientation of a Surface and the Choice of a Side of it 363. The Case of a Piece-Wise Smooth Surface §2. The Area of a Curved Surface 364. Schwarz's Example 365. The Area of a Surface Given by an Explicit Equation 366. The Area of a Surface in the General Case 367. Examples §3. Surface Integrals of the First Type 368. The Definition of a Surface Integral of the First Type 369. The Reduction to an Ordinary Double Integral 370. Mechanical Applications of Surface Integrals of the First Type §4. Surface Integrals of the Second Type 371. The Definition of Surface Integrals of the Second Type 372. The Reduction to an Ordinary Double Integral 373. Stokes's Formula 374. The Application of Stokes's Formula to the Investigation of Curvilinear Integrals in SpaceChapter 23 Triple Integrals §1. A Triple Integral and its Evaluation 375. The Problem of Calculating the Mass of a Solid 376. A Triple Integral and the Conditions for its Existence 377. The Properties of Integrable Functions and Triple Integrals 378. The Evaluation of a Triple Integral 379. Mechanical Applications §2. Ostrogradski's Formula 380. Ostrogradski's Formula 381. Some Examples of Applications of Ostrogradski's Formula §3. Change of Variables in Triple Integrals 382. The Transformation of Space Domains 383. An Expression for Volume in Curvilinear Coordinates 384. A Geometrical Derivation 385. Change of Variables in Triple Integrals 386. Examples 387. Historical Note §4. The Elementary Theory of a Field 388. Scalars and Vectors 389. Scalar and Vector Fields 390. A Derivative in a Given Direction. Gradient 391. The Flow of a Vector Through a Surface 392. Ostrogradski's Formula. Divergence 393. The Circulation of a Vector. Stokes's Formula. Vortex §5. Multiple Integrals 394. The Volume of An m-Dimensional Body and the m-Tuple integral 395. ExamplesChapter 24 Fourier Series §1. Introduction 396. Periodic Values and Harmonic Analysis 397. The Determination of Coefficients by the Euler-Fourier Method 398. Orthogonal Systems of Functions §2. The Expansion of Functions in Fourier Series 399. Statement of the Problem. Dirichlet's Integral 400. A Fundamental Lemma 401. The Principle of Localization 402. The Representation of a Function by Fourier series 403. The Case of a Non-Periodic Function 404. The Case of an Arbitrary Interval 405. An Expansion in Cosines Only, or in Sines Only 406. Examples 407. The Expansion of a Continuous Function in a Series of Trigonometrical Polynomials §3. The Fourier Integral 408. The Fourier Integral as a Limiting Case of a Fourier Series 409. Preliminary Remarks 410. The Representation of a Function by a Fourier Integral 411. Different Forms of Fourier's Formula 412. Fourier Transforms §4. The Closed and Complete Nature of a Trigonometrical System of Functions 413. Mean Approximation to Functions. Extreme Properties of a Fourier Series 414. The Closure of a Trigonometrical System 415. The Completeness of a Trigonometrical System 416. The Generalized Equation of Closure 417. Termwise Integration of a Fourier Series 418. The Geometrical Interpretation §5. An Outline of the History of Trigonometrical Series 419. The Problem of the Vibration of a String 420. D'Alembert's and Euler's Solution 421. Taylor's and D. Bernoulli's Solution 422. The Controversy Concerning the Problem of the Vibration of a String 423. The Expansion of Functions in Trigonometrical Series; The Determination of Coefficient 424. The Proof of the Convergence of Fourier Series and Other Problems 425. Concluding RemarksConclusion an Outline of Further Developments in Mathematical Analysis I. The Theory of Differential Equations II. Variational Calculus III. The Theory of Functions of a Complex Variable IV. The Theory of Integral Equations V. The Theory of Functions of a Real Variable VI. Functional AnalysisIndexOther Titles in the Series