# Mathematical Analysis

### Functions, Limits, Series, Continued Fractions

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Mathematical Analysis: Functions, Limits, Series, Continued Fractions provides an introduction to the differential and integral calculus. This book presents the general problems of the theory of continuous functions of one and several variables, as well as the theory of limiting values for sequences of numbers and vectors. Organized into six chapters, this book begins with an overview of real numbers, the arithmetic linear continuum, limiting values, and functions of one variable. This text then presents the theory of series and practical methods of summation. Other chapters consider the theory of numerical series and series of functions and other analogous processes, particularly infinite continued fractions. This book discusses as well the general problems of the reduction of functions to orthogonal series. The final chapter deals with constants and the most important systems of numbers, including Bernoulli and Euler numbers. This book is a valuable resource for mathematicians, engineers, and research workers.

## Table of Contents

Foreword

Chapter I The Arithmetical Linear Continuum and Functions Defined There

§ 1. Real Numbers and their Representation

1. Real Numbers

2. The Numerical Straight Line

3. p-Adic Systems

4. Sets of Real Numbers

5. Bounded Sets, Upper and Lower Bounds

6. The Theory of Irrational Numbers

§ 2. Functions. Sequences

1. Functions of One Variable

2. Upper and Lower Bounds of a Function

3. Even and Odd Functions

4. Inverse Functions

5. Periodic Functions

6. Functional Equations

7. Numerical Sequences

8. Upper and Lower Bounds of a Sequence

9. Maximum Term of a Sequence

10. Monotonic Sequences

11. Double Sequences

§ 3. Passage to the Limit

1. The Limit Point of a Set

2. The Limit Point and Limit of a Sequence

3. Fundamental Theorems Concerning Limits

4. Some Propositions on Limits

5. Upper and Lower Limits of a Sequence

6. Uniformly Distributed Sequences

7. Recurrent Sequences

8. The Symbols o(αn) and O(αn)

9. Limit of a Function

10. Right and Left Continuity of a Function

11. Continuous and Discontinuous Functions

12. Functional Sequences

13. Uniform Convergence of Functions

14. Convergence in the Mean

15. The Symbols o(x) and O(x)

16. Monotonie Functions

17. Convex Functions

Chapter II n-Dimensional Spaces and Functions Defined There

Introduction

§ 1, n-dimensional Spaces

1. n-Dimensional Coordinate Space

2. n-dimensional Vector Space

3. Scalar Product

4. A Linear System and its Basis

5. Linear Functions

6. Linear Envelope

7. Orthogonal Systems of Vectors

8. Biorthogonal Systems of Vectors

9. The Projection of a Vector on to a Manifold

§ 2. Passage to the Limit, Continuous Functions and Operators

1. Passage to the Limit in n-dimensional Space

2. Series of Vectors

3. Continuous Functions of n Variables

4. Periodic Functions of n Variables. Manifolds of Constancy

5. Passage to the Limit for Linear Envelopes

6. Operators From En into Em

7. Iterative Sequences

8. The Principle of Contraction Mappings

§ 3. Convex Bodies in n-dimensional Space

1. Fundamental Definitions

2. Convex Functions

3. Convex Bodies and the Norm of a Vector

4. Support Hyperplanes

5. Support Functions and Conjugate Spaces

6. Fundamental Theorems on Support Hyperplanes

7. The Connection Between Reciprocal Convex Bodies

8. The Cone. The Tangent Cone

9. Helly's Theorem

10. Linear Operations on Sets

Chapter III Series

Introduction

1. Basic Concepts

2. Some Convergence Tests for Series

§ 1. Numerical Series

1. Alternating Series and Series of Constant Sign

2. Properties of Convergent Series. The Associative Property

3. General Tests for the Convergence of Series of Positive Terms

4. Remainder Term Estimates Corresponding to the Various Convergence Tests

5. Special Tests for the Convergence of Series of Positive Terms. Estimates of the Remainder Term

6. The Convergence of Alternating Series

7. Infinite Products and their Convergence

8. Double Series. Fundamental Concepts and Definitions

9. Some Properties of Double Series

10. Some Convergence Tests for Double Series of Positive Terms. Estimates of Remainder Term

§ 2. Series of Functions

1. Fundamental Properties and Convergence Tests

2. Power Series

3. Operations on Power Series. Taylor Series. Integration and Differentiation of Power Series

4. Complex Series

5. Trigonometric Fourier Series

6. Asymptotic Series

7. Some Methods of Generalized Summation of Divergent Series

§ 3. Methods of Calculating the Sum of a Series

1. Elementary Methods of Exact Summation

2. Summation of Series with the Aid of Functions of a Complex Variable

3. Summation of Series with the Aid of Laplace Transforms

4. Integral Estimations for Finite Sums and Infinite Series

5. Kummer's Transformation

6. Improvement of the Convergence of Series Corresponding to a Given Convergence Test

7. Abel's Transformation

8. A. N. Krylov's Method of Improving the Convergence of Trigonometric Series

9. A. S. Maliyev's Method of Improving the Convergence of Trigonometric Series

Chapter IV Orthogonal Series and Orthogonal Systems

Introduction

§ 1. Orthogonal Systems

1. Orthogonal Systems of Functions Defined at n Points

2. Orthogonal Systems in En(x1, x2, . . . , Xn)

3. The Best Mean Square Approximation

4. Orthogonal Systems of Trigonometric Functions

§ 2. General Properties of Orthogonal and Biorthogonal Systems

1. Orthogonality. Scalar (Inner) Product

2. Orthogonal Systems of Bessel Functions, Haar Functions, Etc.

3. Linear Independence. The Process of Orthogonalization

4. Fourier Coefficients. Closed Systems

5. Fourier Series in a Trigonometric System

6. Biorthogonal Systems of Functions

§ 3. Orthogonal Systems of Polynomials

1. Zeros of Orthogonal Polynomials

2. Recurrence Relations for Orthogonal Polynomials

3. Power Moments. The Expression of Orthogonal Polynomials in Terms of Power Moments

4. The Connection Between Orthogonal Polynomials and Continued Fractions

5. The Conversion of Orthogonal Expansions into a Sequence of Approximating Fractions

6. Orthogonal Polynomials and Quadrature Formulae of the Gaussian Type

7. The Closure of an Orthogonal System of Polynomials

8. Christoffel's Formula. The Convergence of Fourier Series in Orthogonal Polynomials

§ 4. Classical Systems of Orthogonal Polynomials

1. Pearson's Differential Equation

2. The Differential Equations for Corresponding Classes of Orthogonal Polynomials

3. The Expression, by Means of the Weight, of a Polynomial of the nth Degree Belonging to an Orthogonal System of Polynomials

4. The Generating Function of an Orthogonal System of Polynomials with Pearson's Weight

5. Legendre Polynomials

6. Jacobi Polynomials

7. Chebyshev Polynomials of the First Kind

8. Chebyshev Polynomials of the Second Kind

9. Laguerre Polynomials

10. Hermite Polynomials

11. Chebyshev Polynomials, Orthogonal on a Finite System of Points

Chapter V Continued Fractions

Introduction

1. Notation for Continued Fractions. Basic Definitions

2. A Brief Historical Note

§ 1. Continued Fractions and their Fundamental Properties

1. The Evaluation of Convergents. Convergents

2. Transformations of Continued Fractions

3. Contraction and Extension of Continued Fractions

4. The Transformation of a Continued Fraction Resulting from a Theorem of Stolz

5. The Properties of Regular Continued Fractions

6. Equivalent and Corresponding Continued Fractions

7. The Formation of Corresponding Fractions. Viskovatov's Method

8. Appell's Method

§ 2. Fundamental Tests for the Convergence of Continued Fractions

1. The Convergence of Continued Fractions

2. A Necessary and Sufficient Test for the Convergence of a Continued Fraction in Which the Partial Quotients have Positive Terms (Seidel's Test)

3. Tests Sufficient for the Convergence of Continued Fractions in Which the Partial Quotients have Positive Terms

4. First Set of Tests Sufficient for Convergence

5. Tests for the Convergence of Continued Fractions Periodic in the Limit

§ 3. The Expansion of Certain Functions as Continued Fractions

1. Lagrange's Method

2. Fundamental Differential Equation

3. The Expansion of a Power Function as a Continued Fraction

4. The Expansion of a Logarithmic Function as a Continued Fraction

5. The Expansion of an Exponential Function as a Continued Fraction

6. Expansion of the Function Y = arc tan x as a Continued Fraction

7. Expansion of the Function Y = ∫ x 0 dt/1+tk as a Continued Fraction

8. Expansion of tan x and tanh x as Continued Fractions

9. Expansion of Prima's Function as a Continued Fraction

10. Expansion of the Incomplete Gamma Function as a Continued Fraction

11. Thiele's Formula

12. Fractional Approximations for sin x and sinh x

13. Fractional Approximations for cos x and cosh x

14. Fractional Approximation for the Error Function

15. Conversion of Stirling's Series into a Continued Fraction

16. Fractional Approximation for the Gamma Function

17. Fractional Approximation for the Logarithm of the Gamma Function

18. Fractional Approximation for the Derivative of the Logarithm of the Gamma Function

19. Obreshkov's Formula

§ 4. Matrix Methods

1. Extraction of the Square Root by Means of Second-Order Matrices

2. Solution of Quadratic Equations with the Aid of Second-Order Matrices

3. The Connection Between Matrix Methods and the Theory of Continued Fractions

4. The Reduction of Quadratic Surds to Non-Periodic Continued Fractions by Means of Second-Order Matrices with Variable Elements

5. Extraction of the Root of Any Rational Power by Means of Matrices

6. Solution of Cubic Equations by Means of Matrices

7. Recurrent Series. The Bernoulli-Euler Method

8. Connection Between the Bernoulli-Euler Method and Matrix Methods

9. Solution of Higher Degree Equations by Means of Matrices

10. The Idea Behind Jacobi's Algorithm

Chapter VI Some Special Constants and Function

§ 1. Various Constants and Expressions

1. Some Well-Known Constants

2. Some Numerical Expressions

§ 2. Bernoulli and Euler Numbers and Polynomials

1. Bernoulli Numbers and Polynomials

2. Euler Numbers and Polynomials

§ 3. Elementary Piecewise Linear Functions and Delta-Shaped Functions

1. Piecewise Linear Functions

2. The δ(Delta)-Function

§ 4. Elementary Special Functions

1. Elliptic Integrals

2. Integral Functions

3. The Error Function

4. Fresnel Integrals

5. Gamma and Beta Functions of Euler

6. Bessel Functions

Nomenclature

References

Index

Volumes Published in the Series in Pure and Applied Mathematics

## Product details

- No. of pages: 418
- Language: English
- Copyright: © Pergamon 1965
- Published: January 1, 1965
- Imprint: Pergamon
- eBook ISBN: 9781483194363