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The Fundamentals of Mathematical Analysis - 1st Edition - ISBN: 9780080100609, 9781483154138

The Fundamentals of Mathematical Analysis

1st Edition

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Author: G. M. Fikhtengol'ts
Editors: I. N. Sneddon M. Stark S. Ulam
eBook ISBN: 9781483154138
Imprint: Pergamon
Published Date: 1st January 1965
Page Count: 540
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Table of Contents

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 Formula

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 Logarithms

Chapter 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 Theory

Chapter 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 Methods

Chapter 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 Operations

Chapter 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 Matrices

Chapter 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 Problems

Chapter 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 Remarks

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 Note

Chapter 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 Space

Chapter 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 W-Dimensional Body and the W-Tuple Integral

395. Examples

Chapter 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 Series 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 Coefficients

424. The Proof of the Convergence of Fourier Series and Other Problems

425. Concluding Remarks

Conclusion 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 Analysis


Other Titles in the Series


The Fundamentals of Mathematical Analysis, Volume 2 is a continuation of the discussion of the fundamentals of mathematical analysis, specifically on the subject of curvilinear and surface integrals, with emphasis on the difference between the curvilinear and surface ""integrals of first kind"" and ""integrals of second kind.""
The discussions in the book start with an introduction to the elementary concepts of series of numbers, infinite sequences and their limits, and the continuity of the sum of a series. The definition of improper integrals of unbounded functions and that of uniform convergence of integrals are explained. Curvilinear integrals of the first and second kinds are analyzed mathematically. The book then notes the application of surface integrals, through a parametric representation of a surface, and the calculation of the mass of a solid. The text also highlights that Green's formula, which connects a double integral over a plane domain with curvilinear integral along the contour of the domain, has an analogue in Ostrogradski's formula. The periodic values and harmonic analysis such as that found in the operation of a steam engine are analyzed. The volume ends with a note of further developments in mathematical analysis, which is a chronological presentation of important milestones in the history of analysis.
The book is an ideal reference for mathematicians, students, and professors of calculus and advanced mathematics.


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© Pergamon 1965
1st January 1965
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About the Author

G. M. Fikhtengol'ts

About the Editors

I. N. Sneddon

M. Stark

S. Ulam