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*The Applications Of Continued Fractions And Their Generalizations To Problems In Approximation Theory* by A. N. Khovanskii.

# About the book

This book on continued fractions is devoted to certain selected topics in the analytic theory, with particular emphasis on those aspects that deal with rational approximations to functions and with numerical applications and computations. This text contains a tremendous mass of valuable formulas in continued fraction theory. Due to this fact, it can be considered as a useful reference manual for such formulas as well as a text on methods for research in analysis and in computational work.In the first chapter of the present work a short exposition of the analytic theory of continued fractions is given. Problems in the arithmetic theory of continued fractions are not considered in this book.The second chapter is devoted to the continued fraction expansion (by the method of Lagrange) of some well known functions. All expansions given in this chapter are special cases of a general expansion derived at the beginning of the chapter.In the third chapter there is a short consideration of further methods for deriving rational function approximations to functions, leading to a series of approximation formulae for computing certain well known functions.In the fourth chapter are considered the generalized continued fractions proposed by Euler. Examples are quoted showing the possibility of further generalizations of continued fractions which permit the approximate solution of algebraic equations of arbitrary degree.

The book was translated from Russian by Peter Wynn was published in 1963.

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# Contents

## CHAPTER I

Certain Problems in the Theory of Continued Fractions

§ 1. Convergents 1

§ 2. Transformations of Continued fractions 9

§ 3. The Transformation of Series into Continued Fractions 23

§ 4. General Considerations in the Convergence Theory of Continued Fractions 31

§ 5. Convergence Tests for Continued Fractions with Positive Coefficients 42

§ 6. Convergence Tests for Continued Fractions with Arbitrary Coefficients 46

§ 7. Convergence Tests for Continued Reactions which are Periodic in the Limit 58

## CHAPTER II

Continued Fraction Expansions of Certain Functions

§ 1. A Solution of a Certain Riccati Equation with the help of Continued Fractions 76

§ 2. Continued Fraction Expansions. of Binomial Functions 100

§ 3. The Continued Fraction Expansion of $\sqrt[x]{x}$ 109

§ 4. Continued Fraction Expansions of the Natural Logarithm 110

§ 5. Continued Fraction Expansions of $e^{x}$ 112

§ 6. Continued Fraction Expansions of the Inverse trigonometric and Hyperbolic Functions 114

§ 7. Continued Fraction Expansions for $\tan x$ ind $\tanh x$ 122

§ 8. The Continued Fraction Expansion of the Integral $\int_{0}^{x}\frac{dx}{1+x^{k}}$ 125

§ 9. The Solution of the Equations of Boole and Riccati with the help of Continued Fractions 130

§ 10. Continued Fractions and the Hypergeometric Series 133

§ 11. Continued Fraction Expansions of Prym’s Function 142

§ 12. The Continued Fraction Expansion of the Incomplete Gamma-Function 148

## CHAPTER III

Further Methods for Obtaining Rational Function Approximations

§ 1. Obreschkoff’s Formula 151

§ 2. The Derivation of Rational Function approximations to Certain Functions with the Help of Obreschkoff’s Formula 155

§ 3. The Solution of Certain Difference Equations with the Help of Continued Fractions 159

§ 4. The Derivation of Rational Function Approximations by means of Iteration 163

§ 5. Table of Approximate Values of $e^{x}$ 165

§ 6. Table of Approximate Values of $\ln x$ 166

§ 7. Table of Approximate Values of $\tan x$ and $\tanh x$ 167

§ 8. Rational Function Approximations for $\sinh x$ and $\sin x$ 167

§ 9. Rational Function Approximations for cosh x and cos x 171

§ 10. Rational Function Approximations to the Error Function 174

§ 11. The Continued Fraction Expansion of Stirling’: 5 Series 175

§ 12. Rational Function Approximations for $\Gamma(1 + x)$ 177

## CHAPTER IV

Generalized Continued Fractions

§ 1. The Computation of Square Roots with the Help of Matrices of the Second Order 182

§ 2. The Solution of Quadratic Equations with the Help of Matrices of the Second Order 188

$ 3. The Calculation of Cube Roots with the Help of Matrices 194

§ 4. The Calculation of Fourth Roots with the Help of Matrices 196

§ 5. The Calculation of Roots of Arbitrary Rational Order with the Help of a Matrix 198

§ 6. The Solution of Cubic Equations with the Help of Matrices 199

$ 7. The Solution of Equations of Higher Order with the Help of Matrices 201

LITERATURE IN THE RUSSIAN LANGUAGE ON THE GENERAL THEORY OF CONTINUED FRACTIONS 203

REFERENCES 204

SUPPLEMENTARY REFERENCES 210

INDEX 211

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