The Fundamentals of Mathematical Analysis

The Fundamentals of Mathematical Analysis

1st Edition - January 1, 1965

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  • Editor: I. N. Sneddon
  • eBook ISBN: 9781483139074

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The Fundamentals of Mathematical Analysis, Volume 1 is a textbook that provides a systematic and rigorous treatment of the fundamentals of mathematical analysis. Emphasis is placed on the concept of limit which plays a principal role in mathematical analysis. Examples of the application of mathematical analysis to geometry, mechanics, physics, and engineering are given. This volume is comprised of 14 chapters and begins with a discussion on real numbers, their properties and applications, and arithmetical operations over real numbers. The reader is then introduced to the concept of function, important classes of functions, and functions of one variable; the theory of limits and the limit of a function, monotonic functions, and the principle of convergence; and continuous functions of one variable. A systematic account of the differential and integral calculus is then presented, paying particular attention to differentiation of functions of one variable; investigation of the behavior of functions by means of derivatives; functions of several variables; and differentiation of functions of several variables. The remaining chapters focus on the concept of a primitive function (and of an indefinite integral); definite integral; geometric applications of integral and differential calculus. This book is intended for first- and second-year mathematics students.

Table of Contents

  • Volume I


    Chapter 1 Real Numbers

    § 1. The Set of Real Numbers and Its Ordering

    1. Introductory Remarks

    2. Definition of Irrational Number

    3. Ordering of the Set of Real Numbers

    4. Representation of a Real Number By an Infinite Decimal Fraction

    5. Continuity of the Set of Real Numbers

    6. Bounds of Number Sets

    § 2. Arithmetical Operations Over Real Numbers

    7. Definition and Properties of a Sum of Real Numbers

    8. Symmetric Numbers. Absolute Quantity

    9. Definition and Properties of a Product of Real Numbers

    § 3. Further Properties and Applications of Real Numbers

    10. Existence of a Root. Power with a Rational Exponent

    11. Power with an Arbitrary Real Exponent

    12. Logarithms

    13. Measuring Segments

    Chapter 2 Functions of One Variable

    § 1. The Concept of a Function

    14. Variable Quantity

    15. The Domain of Variation of a Variable Quantity

    16. Functional Relation between Variables. Examples

    17. Definition of the Concept of Function

    18. Analytic Method of Prescribing a Function

    19. Graph of a Function

    20. Functions of Positive Integral Argument

    21. Historical Remarks

    § 2. Important Classes of Functions

    22. Elementary Functions

    23. The Concept of the Inverse Function

    24. Inverse Trigonometric Functions

    25. Superposition of Functions. Concluding Remarks

    Chapter 3 Theory of Limits

    § 1. The Limit of a Function

    26. Historical Remarks

    27. Numerical Sequence

    28. Definition of the Limit of a Sequence

    29. Infinitesimal Quantities

    30. Examples

    31. Infinitely Large Quantities

    32. Definition of the Limit of a Function

    33. Another Definition of the Limit of a Function

    34. Examples

    35. One-Sided Limits

    § 2. Theorems on Limits

    36. Properties of Functions of a Positive Integral Argument, Possessing a Finite Limit

    37. Extension to the Case of a Function of an Arbitrary Variable

    38. Passage to the Limit in Equalities and Inequalities

    39. Theorems on Infinitesimals

    40. Arithmetical Operations on Variables

    41. Indefinite Expressions

    42. Extension to the Case of a Function of an Arbitrary Variable

    43. Examples

    § 3. Monotonic Functions

    44. Limit of a Monotonic Function of a Positive Integral Argument

    45. Examples

    46. A Lemma on Imbedded Intervals

    47. The Limit of a Monotonic Function in the General Case

    § 4. The Number e

    48. The Number e Defined as the Limit of a Sequence

    49. Approximate Computation of the Number e

    50. The Basic Formula for the Number e. Natural Logarithms

    § 5. The Principle of Convergence

    51. Partial Sequences

    52. The Condition of Existence of a Finite Limit for a Function of Positive Integral Argument

    53. The Condition of Existence of a Finite Limit for a Function of an Arbitrary Argument

    § 6. Classification of Infinitely Small and Infinitely Large Quantities

    54. Comparison of Infinitesimals

    55. The Scale of Infinitesimals

    56. Equivalent Infinitesimals

    57. Separation of the Principal Part

    58. Problems

    59. Classification of Infinitely Large Quantities

    Chapter 4 Continuous Functions of One Variable

    § 1. Continuity (and Discontinuity) of a Function

    60. Definition of the Continuity of a Function at a Point

    61. Condition of Continuity of a Monotonic Function

    62. Arithmetical Operations over Continuous Functions

    63. Continuity of Elementary Functions

    64. The Superposition of Continuous Functions

    65. Computation of Certain Limits

    66. Power-Exponential Expressions

    67. Classification of Discontinuities. Examples

    § 2. Properties of Continuous Functions

    68. Theorem on the Zeros of a Function

    69. Application to the Solution of Equations

    70. Mean Value Theorem

    71. The Existence of Inverse Functions

    72. Theorem on the Boundedness of a Function

    73. The Greatest and Smallest Values of a Function

    74. The Concept of Uniform Continuity

    75. Theorem on Uniform Continuity

    Chapter 5 Differentiation of Functions of One Variable

    § 1. Derivative of a Function and Its Computation

    76. Problem of Calculating the Velocity of a Moving Point

    77. Problem of Constructing a Tangent to a Curve

    78. Definition of the Derivative

    79. Examples of the Calculation of the Derivative

    80. Derivative of the Inverse Function

    81. Summary of Formula for Derivatives

    82. Formula for the Increment of a Function

    83. Rules for the Calculation of Derivatives

    84. Derivative of a Compound Function

    85. Examples

    86. One-Sided Derivatives

    87. Infinite Derivatives

    88. Further Examples of Exceptional Cases

    § 2. The Differential

    89. Definition of the Differential

    90. The Relation between the Differentiability and the Existence of the Derivative

    91. Fundamental Formula and Rules of Differentiation

    92. Invariance of the Form of the Differential

    93. Differentials as a Source of Approximate Formula

    94. Application of Differentials in Estimating Errors

    § 3. Derivatives and Differentials of Higher Orders

    95. Definition of Derivatives of Higher Orders

    96. General Formula for Derivatives of Arbitrary Order

    97. The Leibniz Formula

    98. Differentials of Higher Orders

    99. Violation of the Invariance of the Form for Differentials of Higher Orders

    Chapter 6 Basic Theorems of Differential Calculus

    § 1. Mean Value Theorems

    100. Fermat's Theorem

    101. Rolle's Theorem

    102. Theorem on Finite Increments

    103. The Limit of the Derivative

    104. Generalized Theorem on Finite Increments

    § 2. Taylor's Formula

    105. Taylor's Formula for a Polynomial

    106. Expansion of an Arbitrary Function

    107. Another Form for the Remainder Term

    108. Application of the Derived Formula to Elementary Functions

    109. Approximate Formula. Examples

    Chapter 7 Investigation of Functions by Means of Derivatives

    § 1. Investigation of the Behavior of Functions

    110. Conditions That a Function May Be Constant

    111. Condition of Monotonicity of a Function

    112. Maxima and Minima; Necessary Conditions

    113. The First Rule

    114. The Second Rule

    115. Construction of the Graph of a Function

    116. Examples

    117. Application of Higher Derivatives

    § 2. The Greatest and the Smallest Values of a Function

    118. Determination of the Greatest and the Smallest Values

    119. Problems

    § 3. Solution of Indeterminate Forms

    120. Indeterminate Forms of the Type 0/0

    121. Indeterminate Forms of the Type ∞/∞

    122. Other Types of Indeterminate Forms

    Chapter 8 Functions of Several Variables

    § 1. Basic Concepts

    123. Functional Dependence between Variables. Examples

    124. Functions of Two Variables and Their Domains of Definition

    125. Arithmetic m-Dimensional Space

    126. Examples of Domains in m-Dimensional Space

    127. General Definition of Open and Closed Domains

    128. Function of m Variables

    129. Limit of a Function of Several Variables

    130. Examples

    131. Repeated Limits

    § 2. Continuous Functions

    132. Continuity and Discontinuities of Functions of Several Variables

    133. Operations on Continuous Functions

    134. Theorem on the Vanishing of a Function

    135. The Bolzano-Weierstrass Lemma

    136. Theorem on the Boundedness of a Function

    137. Uniform Continuity

    Chapter 9 Differentiation of Functions of Several Variables

    § 1. Derivatives and Differentials of Functions of Several Variables

    138. Partial Derivatives

    139. Total Increment of the Function

    140. Derivatives of Compound Functions

    141. Examples

    142. The Total Differential

    143. Invariance of the Form of the (First) Differential

    144. Application of the Total Differential to Approximate Calculations

    145. Homogeneous Functions

    § 2. Derivatives and Differentials of Higher Orders

    146. Derivatives of Higher Orders

    147. Theorems on Mixed Derivatives

    148. Differentials of Higher Orders

    149. Differentials of Compound Functions

    150. The Taylor Formula

    § 3. Extrema, the Greatest and the Smallest Values

    151. Extrema of Functions of Several Variables. Necessary Conditions

    152. Investigation of Stationary Points (for the Case of Two Variables)

    153. The Smallest and the Greatest Values of a Function. Examples

    154. Problems

    Chapter 10 Primitive Function (Indefinite Integral)

    § 1. Indefinite Integral and Simple Methods for Its Evaluation

    155. The Concept of a Primitive Function (and of an Indefinite Integral)

    156. The Integral and the Problem of Determination of Area

    157. Collection of the Basic Integrals

    158. Rules of Integration

    159. Examples

    160. Integration by a Change of Variable

    161. Examples

    162. Integration by Parts

    163. Examples

    § 2. Integration of Rational Expressions

    164. Formulation of the Problem of Integration in Finite Form

    165. Simple Fractions and Their Integration

    166. Integration of Proper Fractions

    167. Ostrogradski's Method for Separating the Rational Part of an Integral

    § 3. Integration of Some Expressions Containing Roots

    168. Integration of Expressions of the Form R[x,m√(αx+ß/γx+δ)]

    169. Integration of Binomial Differentials

    170. Integration of Expressions of the Form R[x,√(ax2+bx+c)]. Euler's Substitution

    § 4. Integration of Expressions Containing Trigonometric and Exponential Functions

    171. Integration of the Differentials R(Sin x, Cos x)dx

    111. Survey of Other Cases

    § 5. Elliptic Integrals

    173. Definitions

    174. Reduction to the Canonical Form

    Chapter 11 Definite Integral

    § 1. Definition and Conditions for the Existence of a Definite Integral

    175. Another Formulation of the Area Problem

    176. Definition

    177. Darboux's Sums

    178. Condition for the Existence of the Integral

    179. Classes of Integrable Functions

    § 2. Properties of Definite Integrals

    180. Integrals over an Oriented Interval

    181. Properties Expressed by Equalities

    182. Properties Expressed by Inequalities

    183. Definite Integral as a Function of the Upper Limit

    § 3. Evaluation and Transformation of Definite Integrals

    184. Evaluation Using Integral Sums

    185. The Fundamental Formula of Integral Calculus

    186. The Formula for the Change of Variable in a Definite Integral

    187. Integration By Parts in a Definite Integral

    188. Wallis's Formula

    § 4. Approximate Evaluation of Integrals

    189. The Trapezium Formula

    190. Parabolic Formula

    191. Remainder Term for the Approximate Formula

    192. Example

    Chapter 12 Geometric and Mechanical Applications of the Integral Calculus

    § 1. Areas and Volumes

    193. Definition of the Concept of Area. Quadrable Domains

    194. The Additive Property of Area

    195. Area as a Limit

    196. An Integral Expression for Area

    197. Definition of the Concept of Volume and Its Properties

    198. Integral Expression for the Volume

    § 2. Length of Arc

    199. Definition of the Concept of the Length of an Arc

    200. Lemmas

    201. Integral Expression for the Length of an Arc

    202. Variable Arc and Its Differential

    203. Length of the Arc of a Spatial Curve

    § 3. Computation of Mechanical and Physical Quantities

    204. Applications of Definite Integrals

    205. The Area of a Surface of Revolution

    206. Calculation of Static Moments and Center of Mass of a Curve

    207. Determination of Static Moments and Center of Mass of a Plane Figure

    208. Mechanical Work

    Chapter 13 Some Geometric Applications of the Differential Calculus

    § 1. The Tangent and the Tangent Plane

    209. Analytic Representation of Plane Curves

    210. Tangent to a Plane Curve

    211. Positive Direction of the Tangent

    212. The Case of a Spatial Curve

    213. The Tangent Plane to a Surface

    § 2. Curvature of a Plane Curve

    214. The Direction of Concavity, Points of Inflection

    215. The Concept of Curvature

    216. The Circle of Curvature and Radius of Curvature

    Chapter 14 Historical Survey of the Development of the Fundamental Concepts of Mathematical Analysis

    § 1. Early History of the Differential and Integral Calculus

    217. Seventeenth Century and the Analysis of Infinitesimals

    218. The Method of Indivisibles

    219. Further Development of the Science of Indivisibles

    220. Determination of the Greatest and Smallest Quantities; Construction of Tangents

    221. Construction of Tangents By Means of Kinematic Considerations

    222. Mutual Invertibility of the Problems of Construction of Tangent and Squaring

    223. Survey of the Foregoing Achievements

    § 2. Isaac Newton (1642-1727)

    224. The Calculus of Fluxions

    225. The Calculus Inverse to the Calculus of Fluxions; Squaring

    226. Newton's Principles and the Origin of the Theory of Limits

    227. Problems of Foundations in Newton's Works

    § 3. Gottfried Wilhelm Leibniz (1646-1716)

    228. First Steps in Creating the New Calculus

    229. The First Published Work on Differential Calculus

    230. The First Published Paper on Integral Calculus

    231. Further Works of Leibniz. Creation of a School

    232. Problems of Foundation in Leibniz's Works

    233. Postscript


    Other Titles in the Series

    Volume II

    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. 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 m-Dimensional Body and the m-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 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

    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

Product details

  • No. of pages: 520
  • Language: English
  • Copyright: © Pergamon 1965
  • Published: January 1, 1965
  • Imprint: Pergamon
  • eBook ISBN: 9781483139074

About the Editor

I. N. Sneddon

About the Author

G. M. Fikhtengol'ts

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