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 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

Index

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