Fundamental textbook on mathematical analysis, which went through many editions and was translated into a number of foreign languages, differs, on the one hand, in the systematic and rigorous presentation, and on the other, in plain language, detailed explanations and numerous examples illustrating the theory.
The course is intended for students of universities, pedagogical and technical universities and has been used for a long time in various educational institutions as one of the main teaching aids. It allows the student not only to master the theoretical material, but also to acquire the most important practical skills. The course is highly regarded by mathematicians as a unique collection of various facts of analysis, some of which cannot be found in other books in Russian.
The first edition appeared in 1948.
EDITOR'S FOREWORD
Course in differential and integral calculus Grigory Mikhailovich Fikhtengolts is an outstanding work of scientific and pedagogical literature, which has gone through many editions and has been translated into a number of foreign languages. The course is unparalleled in terms of the scope covered actual material, the number of various applications of general theorems in geometry, algebra, mechanics, physics and technology. Many well-known modern mathematicians note that it was G. M. Fikhtengolz's Course that instilled in them in student years taste and love for mathematical analysis, gave the first clear understanding of this subject.
In the 50 years that have passed since the publication of the first edition of the Course, its text has practically not become outdated and in this moment can still be used and used by students of universities, as well as various technical and pedagogical universities as one of the main textbooks on mathematical analysis and the course of higher mathematics. Moreover, despite the emergence of new good textbooks During its existence, the readership of the GM Fikhtengolts Course has only expanded and now includes students from a number of physics and mathematics lyceums, students of advanced mathematical qualification courses for engineers.
High level demand for the course is explained by its unique features. Basic theoretical material included in the Course is a classic part of the modern mathematical analysis, which was finally formed by the beginning of the 20th century (does not contain measure theory and general theory sets). This part of the analysis is taught in the first two courses of universities and is included (in whole or in large part) in the programs of all technical and pedagogical universities. Volume I of the Course includes the differential calculus of one and several real variables and its main applications, Volume II is devoted to the theory of the Riemann integral and the theory of series, Volume III - multiple, curvilinear and surface integrals, the Stieltjes integral, series and the Fourier transform.
Great amount examples and applications, as a rule, very interesting, some of which cannot be found in other literature in Russian, is one of the main features of the Course, already mentioned above.
Another essential feature is the availability, detail and thoroughness of the presentation of the material. A significant volume of the Course does not become a hindrance to its assimilation. On the contrary, it enables the author to pay sufficient attention to the motivations for new definitions and problem statements, detailed and thorough proofs of the main theorems, and many other aspects that make it easier for the reader to understand the subject. In general, the problem of combining clarity and rigor of presentation (the absence of the latter simply leads to a distortion mathematical facts) is well known to any teacher. Huge pedagogical skill Grigory Mikhailovich allows him throughout the course to give many examples of solving this problem; along with other circumstances, this makes the Course an indispensable model for a novice lecturer and an object of research for specialists in the methodology of teaching higher mathematics.
Another feature of the Course is the very slight use of any elements of set theory (including notation). At the same time, the full rigor of the presentation is preserved; in general, like 50 years ago, this approach makes it easier for a significant part of the readership to initially master the subject.
In the new edition of the Course by G. M. Fikhtengolts, offered to the reader's attention, typographical errors found in a number of previous editions have been eliminated. In addition, the publication is provided with brief comments relating to those places in the text (very few), when working with which the reader may experience certain inconveniences; notes are made, in particular, in cases where the term or turn of speech used by the author differs in some way from the most common at present. The responsibility for the content of the notes lies entirely with the editor of the publication.
The editor is deeply grateful to Professor B. M. Makarov, who read the texts of all the notes and expressed a number of valuable opinions. I would also like to thank all the staff of the Department of Mathematical Analysis of the Faculty of Mathematics and Mechanics of the St. state university who discussed with the author of these lines various issues related to the texts of previous editions and the idea of a new edition of the Course.
The editors would like to thank in advance all readers who wish to further improve the quality of the publication with their comments.
A. A. Florinsky
Fikhtengolts G.M. (2003) Course of differential and integral calculus. T.1.
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CHAPTER EIGHT. DERIVATIVE FUNCTION (INDETERMINATE INTEGRAL)
§ 1. Indefinite integral and the simplest methods of its calculation
263. Concept antiderivative function(and indefinite integral)
264. The Integral and the Area Problem
265. Table of basic integrals
266. The Simplest Integration Rules
267. Examples
268. Integration by Change of Variable
269. Examples
270. Integration by parts
271. Examples
§ 2. Integration rational expressions
272. Statement of the problem of integration in final form
273. simple fractions and their integration
274. Decay proper fractions into simple
275. Determination of coefficients. Integration of proper fractions
276. Separation of the rational part of the integral
277. Examples
§ 3. Integration of some expressions containing radicals
278. Integration of expressions
279. Integration of binomial differentials. Examples
280. Reduction Formulas
281. Integration of expressions. Euler substitutions
282. Geometric treatment of Euler substitutions
283. Examples
284. Other Methods of Calculation
285. Examples
§ 4. Integration of expressions containing trigonometric and exponential functions
286. Integration of Differentials R(sin x, cos x)
287. Integration of expressions
288. Examples
289. Review of other cases
§ 5. Elliptic integrals
290. General remarks and definitions
291. Auxiliary transformations
292. Reduction to canonical form
293. Elliptic integrals of the 1st, 2nd and 3rd kind
CHAPTER NINE. DEFINITION INTEGRAL
§ 1. Definition and conditions for the existence of a definite integral
294. Another approach to the area problem
295. Definition
296. Darboux sums
297. The Condition for the Existence of an Integral
298. Classes of Integrable Functions
299. Properties of Integrable Functions
300. Examples and additions
301. Lower and Upper Integrals as Limits
§ 2. Properties of definite integrals
302. Integral over an oriented interval
303. Properties expressed by equalities
304. Properties Expressed by Inequalities PO
305. Definite Integral as a Function of the Upper Limit
306. Second Mean Value Theorem
§ 3. Calculation and transformation of definite integrals
307 Calculation with integral sums
308. Basic Formula of Integral Calculus
309. Examples
310. Another conclusion basic formula
311. Reduction formulas
312. Examples
313. The formula for the change of variable in a definite integral
314. Examples
315. Gauss formula. Landen transform
316. Another derivation of the change of variable formula
§ 4. Some applications of definite integrals
317. Wallis Formula
318. Taylor formula with an additional term
319. Transcendence of the number e
320. Legendre Polynomials
321. Integral inequalities
§ 5. Approximate calculation of integrals
322. Statement of the problem. Formulas for rectangles and trapezoids
323 Parabolic Interpolation
324. Splitting the Interval of Integration
325. Additional term of the formula of rectangles
326. Additional term of the trapezoid formula
327. Additional term of Simpson's formula
328. Examples
CHAPTER TEN. APPLICATIONS OF THE INTEGRAL CALCULUS TO GEOMETRY, MECHANICS AND PHYSICS
§ 1. Curve length
329 Calculating the Length of a Curve
330. Another approach to the definition of the concept of the length of a curve and its calculation
331. Examples
332. natural equation flat curve
333. Examples
334. Arc Length of a Space Curve
§ 2. Areas and volumes
335. Definition of the concept of area. Additivity property
336. Area as a Limit
337. Classes of squaring regions
338. Expression of area by integral
339. Examples
340. Definition of the concept of volume. Its properties
341. Classes of bodies having volumes
342. Expression of Volume by an Integral
343. Examples
344. Surface area of rotation
345. Examples
346. Square cylindrical surface
347. Examples
§ 3. Calculation of mechanical and physical quantities
348. Scheme of Application of a Definite Integral
349. Finding the Static Moments and the Center of Gravity of a Curve
350. Examples
351. Finding static moments and center of gravity flat figure
352. Examples
353. mechanical work
354. Examples
355. The work of the friction force in a flat heel
356. Problems for the summation of infinitesimal elements
§ 4. The simplest differential equations
357. Basic concepts. First order equations
358. Equations of the first degree with respect to the derivative. Separation of variables
359. Tasks
360. Compilation notes differential equations
361. Tasks
CHAPTER ELEVEN. ENDLESS ROWS WITH PERMANENT MEMBERS
§ 1. Introduction
362. Basic concepts
363. Examples
364. Fundamental Theorems
§ 2. Convergence of positive series
365. Condition for the Convergence of a Positive Series
366. Series Comparison Theorems
367. Examples
368. Signs of Cauchy and D'Alembert
369. Sign of Raabe
370. Examples
371. Sign of Kummer
372. Gauss sign
373. Integral sign of Maclaurin-Cauchy
374. Sign of Ermakov
375. Additions
§ 3. Convergence of arbitrary series
376. General Condition for the Convergence of a Series
377. Absolute Convergence
378. Examples
379. Power Series, Its Interval of Convergence
380. Expression of the radius of convergence in terms of coefficients
381. alternating series
382. Examples
383. Abel Transform
384. Signs of Abel and Dirichlet
385. Examples
§ 4. Properties of convergent series
386. associative property
387. Commutative property of absolutely convergent series
388. The Case of Nonabsolutely Convergent Series
389. Multiplication of rows
390. Examples
391. General theorem from the theory of limits
392. Further theorems on the multiplication of series
§ 5. Repeated and double rows
393. Repeated rows
394. Double rows
395. Examples
396 Power series with two variables; region of convergence
397. Examples
398. Multiple rows
§ 6. Infinite products
399. Basic concepts
400. Examples
401. Basic theorems. Relationship with rows
402. Examples
§ 7. Expansions of elementary functions
403. Expansion of a function in a power series; Taylor series
404. Expansion in a series of exponential, basic trigonometric functions, etc.
405. Logarithmic Series
406. Stirling formula
407. Binomial Series
408. Decomposition of sine and cosine into infinite products
§ 8. Approximate calculations with the help of series. Series conversion
409. General remarks
410. Calculating the number of tt
411. Calculating Logarithms
412. Calculating Roots
413. Euler Series Transformation
414. Examples
415. Kummer's Transformation
416. Markov Transform
§ 9. Summation of divergent series
417. Introduction
418. Power Series Method
419. Tauber's theorem
420. Method of Arithmetic Averages
421. Relationship between Poisson-Abel and Cesaro methods
422. Hardy-Landau theorem
423. Application of generalized summation to multiplication of series
424. Other methods of generalized summation of series
425. Examples
426. General class linear regular summation methods
CHAPTER TWELVE. FUNCTIONAL SEQUENCES AND SERIES
§ 1. Uniform convergence
427. Introductory remarks
428. Uniform and non-uniform convergence
429. Condition for uniform convergence
430. Criteria for Uniform Convergence of Series
§ 2. Functional properties series sums
431. Continuity of the sum of a series
432. A remark on quasi-uniform convergence
433. Transition to the limit term by term
434. Termwise Integration of Series
435. Term Differentiation of Series
436. Sequence Point of View
437. Continuity of the sum of a power series
438. Integration and differentiation of power series
§ 3. Applications
439. Examples on the continuity of the sum of a series and on the passage to the limit term by term
440. Examples for term-by-term integration of series
441. Examples for term-by-term differentiation of series
442. Method successive approximations in the theory of implicit functions
443. Analytical definition trigonometric functions
444. An example of a continuous function without a derivative
§ four. additional information about power series
445. Actions on power series
446. Substituting a row into a row
447. Examples
448. Division of power series
449. Bernoulli numbers and expansions in which they occur
450. Solving Equations in Series
451. Power series inversion
452. Lagrange series
§ 5. Elementary Functions complex variable
453. Complex numbers
454. Complex variant and its limit
455. Functions of a Complex Variable
456. Power series
457. Exponential function
458. logarithmic function
459. Trigonometric functions and their reverse
460. Power function
461. Examples
§ 6. Enveloping and asymptotic series. Euler-Maclaurin formula
462. Examples
463. Definitions
464. Basic Properties of Asymptotic Expansions
465. Derivation of the Euler-Maclaurin Formula
466. Study of an additional member
467. Examples of Calculations Using the Euler-Maclaurin Formula
468. Another form of the Euler-Maclaurin formula
469. Sterling's Formula and Series
CHAPTER THIRTEEN. Improper integrals
§ 1. Improper integrals with infinite limits
470. Definition of integrals with infinite limits
471. Application of the basic formula of integral calculus
472. Examples
473. Analogy with series. The simplest theorems
474. Convergence of the Integral in the Case positive function
475. Convergence of the Integral in the General Case
476. Signs of Abel and Dirichlet
477. Reducing an Improper Integral to an Infinite Series
478. Examples
§ 2. Improper integrals of unbounded functions
479. Definition of Integrals of Unbounded Functions
480. Remark on singular points
481. Application of the basic formula of integral calculus. Examples
482. Conditions and signs of the existence of an integral
483. Examples
484. Principal Values of Improper Integrals
485. A Remark on Generalized Values of Divergent Integrals
§ 3. Properties and transformation of improper integrals
486. The Simplest Properties
487. Mean Value Theorems
488 Integration by Parts in the Case of Improper Integrals
489. Examples
490. Change of Variables in improper integrals
491. Examples
§ 4. Special methods for calculating improper integrals
492. Some Remarkable Integrals
493. Calculation of improper integrals with the help of integral sums. The case of integrals with finite limits
494. The case of integrals with endless limit
495 Frullani Integrals
496. Integrals of rational functions between endless limits
497. Mixed examples and exercises
§ 5. Approximate calculation of improper integrals
498. Integrals with finite limits; highlighting features
499. Examples
500. Remark on Approximate Calculation of Eigenintegrals
501. Approximate calculation of improper integrals with an infinite limit
502. Use of asymptotic expansions
CHAPTER FOURTEEN. INTEGRALS DEPENDING ON A PARAMETER
§ one. elementary theory
503. Statement of the problem
504. Uniform striving for limit function
505. Permutation of two passages to the limit
506. Limit transition under the integral sign
507. Differentiation under the Integral Sign
508. Integration under the integral sign
509. The Case When And The Limits Of The Integral Depend On The Parameter
510. Introduction of a multiplier depending only on x
511. Examples
512. Gaussian proof of the fundamental theorem of algebra
§ 2. Uniform convergence of integrals
513. Definition of uniform convergence of integrals
514. Condition for uniform convergence. Relationship with rows
515. Sufficient signs uniform convergence
516. Another case of uniform convergence
517. Examples
§ 3. Use of uniform convergence of integrals
518. Passing to the limit under the integral sign
519. Examples
520. Continuity and differentiability of an integral with respect to a parameter
521. Integration over a parameter
522. Application to the calculation of certain integrals
523. Examples for Differentiation under the Integral Sign
524. Examples for integration under the integral sign
§ 4. Additions
525. Arzel's Lemma
526. Passing to the limit under the integral sign
527. Differentiation under the Integral Sign
528. Integration under the integral sign
§ 5. Euler integrals
529. Euler integral of the first kind
530. Euler integral of the second kind
531. The Simplest Properties of the Function Γ
532. Unambiguous definition function Γ by its properties
533. Other functional characteristic G functions
534. Examples
535. The logarithmic derivative of the function Г
536. The multiplication theorem for the function Г
537. Some expansions into series and products
538. Examples and additions
539. Calculation of certain definite integrals
540. Stirling formula 9
541 Calculating the Euler Constant
542. Compiling a table decimal logarithms G functions
CHAPTER FIFTEEN. CURVILINEAR INTEGRALS. Stieltjes integral
§ 1. Curvilinear integrals of the first type 11
543. Definition of a Curvilinear Integral of the First Type 11
544. Reduction to the ordinary definite integral 13
545. Examples 15
§ 2. Curvilinear integrals of the second type 20
546. Definition of Curvilinear Integrals of the Second Type 20
547. Existence and calculation of a curvilinear integral of the second type
548. The case of a closed circuit. Plane orientation 25
549. Examples 27
550. Approximation using an integral taken over a broken line 30
551 Calculating Areas Using Curvilinear Integrals 32
552. Examples 35
553. Relationship between Curvilinear Integrals of Both Types 38
554. Physical problems 40 § 3. Conditions for the independence of the curvilinear integral from the path 45
555. Statement of the problem, connection with the question of the exact differential 45
556. Derivation of an integral independent of the path 46
557. Calculation of the Curvilinear Integral through the Antiderivative 49
558. Test for Exact Differential and Finding the Antiderivative in the Case of a Rectangular Domain
559. Generalization to the case of an arbitrary region 52
560. Final results 55
561 Closed loop integrals 56
562. The Case of a Nonsimply Connected Region or the Presence of Singular Points 57
563. Gauss integral 62
564. Three-Dimensional Case 64
565. Examples 67
566. Annex to physical tasks 71
§ 4. Functions with limited variation 74
567. Function definition with limited change 74
568. Classes of Functions with Limited Variation 76
569. Properties of Functions with Limited Variation 79
570. Criteria for functions with limited change 82
571. Continuous functions with limited change 84
572 Rectifiable Curves 87
§ 5. The Stieltjes integral 89
573. Definition of the Stieltjes integral 89
574. General terms existence of the Stieltjes integral 91
575. Classes of cases of the existence of the Stieltjes integral 92
576 Properties of the Stieltjes Integral 95
577. Integration by parts 97
578 Reduction of the Stieltjes integral to the Riemann integral 98
579 Calculation of Stieltjes Integrals 100
580. Examples 104
581. Geometric illustration of the Stieltjes integral 111
582. Mean Theorem, Estimates 112
583 Passing to the limit under the sign of the Stieltjes integral 114
584. Examples and additions 115
585. Reduction of a Curvilinear Integral of the Second Type to a Stieltjes Integral
CHAPTER SIXTEEN. DOUBLE INTEGRALS
§ 1. Definition and elementary properties of the double integral 122
586. The problem of the volume of a cylindrical bar 122
587. Reduction of a double integral to an iterated one 123
588. Definition of the double integral 125
589. Conditions for the existence of a double integral 127
590 Classes of Integrable Functions 128
591. Lower and Upper Integrals as Limits 130
592. Properties of integrable functions and double integrals 131
593. Integral as an additive function of a region; region differentiation
§ 2. Calculation of the double integral 137
594. Reduction of a double integral to an iterated one in the case of a rectangular region
595. Examples 141
596. Reduction of a double integral to an iterated one in the case of a curvilinear region
597. Examples 152
598. Mechanical applications 165
599. Examples 167
§ 3. Green's Formula 174
600. Derivation of Green's Formula 174
601. Application of Green's Formula to the Study of Curvilinear Integrals
602. Examples and additions 179
§ 4. Change of variables in the double integral 182
603. Transforming Flat Areas 182
604. Examples 184
605. Expression of area in curvilinear coordinates 189
606. Additional remarks 192
607. Geometric derivation 194
608. Examples 196
609 Change of Variables in Double Integrals 204
610. Analogy with simple integral. Integral over oriented area
611. Examples 207
§ 5. Improper double integrals 214
612. Integrals extended to an unbounded region 214
613. The theorem on the absolute convergence of an improper double integral
614. Reduction of a double integral to an iterated one 219
615. Integrals of Unbounded Functions 221
616 Change of Variables in Improper Integrals 223
617. Examples 225
CHAPTER SEVENTEEN. SURFACE AREA. SURFACE INTEGRALS
§ 1. Two-sided surfaces 241
618. Surface side 241
617. Examples 243
620. Orientation of surfaces and space 244
621. Choosing a sign in formulas for direction cosines of the normal 246
622. The Case of a Piecewise Smooth Surface 247
§ 2. Area of a curved surface 248
623. Schwartz Example 248
624. Determining the area of a curved surface 251
625. Remark 252
626. Existence of surface area and its calculation 253
627. Approach through inscribed polyhedral surfaces 258
628. Special cases determining the area 259
629. Examples 260
§ 3. Surface integrals of the first type 274
630. Definition of a Surface Integral of the First Type 274
631. Reduction to the ordinary double integral 275
632. Mechanical applications of surface integrals of the first type 277
633. Examples 279
§ 4. Surface integrals of the second type 285
634. Definition of a surface integral of the second type 285
635. The Simplest Special Cases 287
636. General case 290
637. Detail of proof 292
638. Expression of body volume surface integral 293
639. Stokes formula 297
640. Examples 299
641. Application of the Stokes Formula to the Study of Curvilinear Integrals in Space
CHAPTER EIGHTEEN. TRIPLE AND MULTIPLE INTEGRALS
§ 1. The triple integral and its calculation 308
642. The problem of calculating the mass of a body 308
643. Triple Integral and Conditions for Its Existence 309
644. Properties of integrable functions and triple integrals 310
645. Evaluation of the Triple Integral Extended to a Parallelepiped
646. Calculation of the triple integral over any area 314
647 Improper Triple Integrals 315
648. Examples 316
649. Mechanical applications 323
650. Examples 325
§ 2. Gauss-Ostrogradsky formula 333
651. Ostrogradsky's Formula 333
652. Application of the Ostrogradsky formula to the study of surface integrals
653 Gauss integral 336
654. Examples 338
§ 3. Change of variables in triple integrals 342
655. Transformation of spaces and curvilinear coordinates 342
656. Examples 343
657 Expressing Volume in Curvilinear Coordinates 345
658. Additional remarks 348
659. Geometric derivation 349
660. Examples 350
661 Change of Variables in Triple Integrals 358
662. Examples 359
663. Attraction from the side of the body and the potential for inner point 364
§ 4. Elements of vector analysis 366
664. Scalars and Vectors 366
665. Scalar and vector fields 367
666. Gradient 368
667 Vector flow through a surface 370
668. Ostrogradsky's formula. Divergence 371
669. Vector circulation. Stokes formula. Whirlwind 372
670. Special fields 374
671. Inverse problem vector analysis 378
672. Applications 378
§ 5. Multiple integrals 384
673. The Problem of Attraction and Potential of Two Bodies 384
674. Volume of an n-dimensional body, n-fold integral 386
675 Change of variables in n-fold integral 388
676. Examples 391
CHAPTER NINETEEN. FOURIER SERIES
§ 1 Introduction 414
677 Periodic Quantities and Harmonic Analysis 414
678. Determination of Coefficients by the Euler-Fourier Method 417
679. Orthogonal systems of functions 419
680. Trigonometric Interpolation 424
§ 2. Fourier expansion of functions 427
681. Statement of the question. Dirichlet integral 427
682. First main lemma 429
683. The principle of localization 432
684. Dini and Lipschitz Tests for the Convergence of Fourier Series 433
685. Second Main Lemma 436
686. Sign of Dirichlet-Jordan 438
687. Case non-periodic function 440
688. The Case of an Arbitrary Interval 441
689. Expansions only in cosines or only in sines 442
690. Examples 446
691. Decomposition of In T(x) 461
§ 3. Additions 463
692. Series with Decreasing Coefficients 463
693. Summation of trigonometric series using analytic functions complex variable
694. Examples 472
695. complex form Fourier series 477
696. Conjugated series 480
697 Multiple Fourier Series 483
§ 4. The nature of the convergence of Fourier series 484
698. Some additions to the main lemmas 484
699. Tests for uniform convergence of Fourier series 487
700 Behavior of the Fourier series near the discontinuity point; special case 490
701. The Case of an Arbitrary Function 495
702. Singularities of Fourier series; preliminary remarks 497
703. Construction of singularities 500
§ 5. An estimate of the remainder depending on the differential properties of a function 502
704. Connection between the Fourier coefficients of a function and its derivatives 502
705. Appreciation partial amount in case of limited function 503
706. Estimate of the remainder in the case of a function with limited k-th derivative 505
707. The case of a function having kth derivative with limited change
708. Influence of discontinuities of a function and its derivatives on the order of smallness of the Fourier coefficients
709. The case of a function defined in the interval 514
710. Method of extracting features 516
§ 6. Fourier integral 524
711. The Fourier Integral as the Limiting Case of the Fourier Series 524
712. Preliminary remarks 526
713. Sufficient signs 527
714. Modification of the basic assumption 529
715. Different kinds Fourier formulas 532
716. Fourier Transform 534
717. Some Properties of Fourier Transforms 537
718. Examples and additions 538
719. The Case of a Function of Two Variables 545
Section 7 Appendices 547
720. Expression of the eccentric anomaly of a planet in terms of its mean anomaly
721. The Problem of the Vibration of a String 549
722. The Problem of Heat Propagation in a Finite Rod 553
723. The Case of an Infinite Rod 557
724. Modification of limit conditions 559
725. Distribution of Heat in a Round Plate 561
726 Practical Harmonic Analysis Scheme for twelve ordinates
727. Examples 565
728. Scheme for twenty-four ordinates 569
729. Examples 570
730. Comparison of approximate and exact values Fourier coefficients
CHAPTER TWENTY. FOURIER SERIES (continued)
§ 1. Operations on Fourier series. Completeness and closedness 574
731. Term-by-Term Integration of the Fourier Series 574
732. Term Differentiation of the Fourier Series 577
733. Completeness trigonometric system 578
734. Uniform approximation of functions. Weierstrass' theorems 580
735. Approximation of functions on the average. Extremal properties of segments of the Fourier series
736. Closedness of the trigonometric system. Lyapunov's theorem 586
737. Generalized closure equation 589
738. Multiplication of Fourier Series 592
739. Some Applications of the Equation of Closure 593
§ 2. Application of generalized summation methods to Fourier series 599
740. Main Lemma 599
741. Poisson-Abel Summation of Fourier Series 601
742. Solution of the Dirichlet problem for a circle 605
743. Summation of Fourier series by the Ces'aro-Fejér method 607
744. Some applications of the generalized summation of Fourier series 609
745. Term Differentiation of Fourier Series 611
§ 3. Uniqueness trigonometric expansion functions 613
746. Auxiliary Propositions on Generalized Derivatives 613
747. Riemann's method of summation of trigonometric series 616
748. Lemma on the coefficients of a convergent series 620
749. Uniqueness of the trigonometric expansion 621
750. Final theorems on Fourier series 623
751. Generalization 626
ADDITION. GENERAL POINT OF VIEW ON THE LIMIT
752. Different Kinds of Limits Encountered in Analysis 631
753. Ordered Sets (Properly) 632
754. Ordered Sets (in a Generalized Sense) 633
755. An ordered variable and its limit 636
756. Examples 637
757. A remark about the limit of a function 639
758. Extension of the theory of limits 640
759. Equally Ordered Variables 643
760 Ordering with Numeric Parameter 644
761. Reduction to variant 645
762. Largest and Smallest Limits of an Ordered Variable 647
