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Continued Fractions of the Form (1.6) with Definite Integrals of the Form

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Continued Fractions of the Form (1.6) with Definite Integrals of the Form V CONTINUED FRACTIONS AND THEIR APPLICATION IN THE COMPUTATION OF DEFINITE RIEMANN INTEGRALS by STANLEY MAX COMPTON, B.S. A THESIS IN MATHEMATICS Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Approved August, 1973 ^ ^^- ^J 1973 ACKNOWLEDGMENTS I would like to express my sincere appreciation to Professor Vadim Komkov for his guidance with respect to the development of this thesis, and to Professors Charles N. Kellogg and Robert A. Moreland for their helpful criticisms. I would also like to thank Mr. Joel R, Wilson for his inspirational interest in this paper, and Mrs. Anna F. Kirk for her fine editing and typing. 11 CONTENTS ACK^^OWLEDGMENTS ii LIST OF TABLES iv I. INTRODUCTION AND HISTORICAL DEVELOPMENT ..... 1 Introduction 1 Historical Development 2 II. INTRODUCTORY CONCEPTS 6 Definitions and Notation 6 Successive Convergents 7 Convergence 11 Continued Fractions and Power Series 14 III. CONTINUED FRACTION EXPANSIONS 17 Introductory Examples 17 Examples of Continued Fraction Expansions for Certain Functions 23 IV. THE QUOTIENT-DIFFERENCE ALGORITHM 25 Introduction 25 Hankel Determinants 25 The Quotient-Difference Algorithm 26 The Quotient-Difference Algorithm and Continued Fractions 35 Continued Fractions and Definite Riemann Integrals 43 BIBLIOGRAPHY 51 111 LIST OF TABLES Tanle A 30 Table B 31 Table C 34 Table D 39 Table E 42 Table F 44 Table G 46 Table H 49 IV CHAPTER I INTRODUCTION AND HISTORICAL DEVELOPMENT Introduction Almost unknown to the Western student of modern mathematics, the subject of continued fractions plays an extensive, if not vital, role in analytic theory as well as arithmetic theory. Powerful applications involving the use of continued fractions exist with respect to the theory of equations, orthogonal polynomials, power series, infinite matrices and quadratic forms in infinitely many variables, definite integrals, the moment problem, analytic functions, and the summation of divergent series. As an elementary example, let us suppose we are going to solve the quadratic equation (1.1) x^ - 3x - 1 = 0. We might begin by dividing equation (1.1) by x and write (1.2) X = 3 + ^. Then we might replace x in the right-hand side of equation (1.2) with its equal, namely 3 + —, to obtain .TV (1.3) x = 3 + ^. 3 + - X With repeated substitutions for x in the right-hand side of equation (1.3) we obtain (1.4) x = 3 -»- 3 + I 1 3 +—^ 3 + - X Now it is clear that this procedure will never yield an exact solution of equation (1.1). However, if this procedure were to be truncated as in expression (1.4), and the results numerically computed, we would obtain a good approximation to the positive root of equation (1.1). In fact, the accuracy of this approximation varies as the number of terms which remain after the truncation has been performed. In Chapter II we shall see that an expansion like the one given in expression (1.4) is called a continued fraction. We shall also see that there is a very eloquent theory associated with such expansions. Historical Development Perhaps the first major step in the creation of an analytic theory for continued fractions came with T. J. Stieltjes* [24] "Recherches sur les Fractions Continues" in 1894. In this paper Stieltjes developed the funda­ mental function theory and integral theory necessary for a complete treatment of an important class of continued fractions. During the years of 1889 through 1890 Stieltjes published several examples of continued fraction expansions of definite integrals of the form ^i^^t. X (z + t) where f(t) > 0. The continued fractions associated with these integrals are of the form (1.5) ^ , z + C2 1 + — ^ z + 1 + where c. > 0, i = 1, 2, 3, ... Stieltjes later showed that the continued fraction (1.5) could be transformed into (1.6) ^1 z + a, T- ^ "^ ^2 " z -H a- - •. where the a. and b. are positive functions of the c. 11 1 In 1903 E. B. Van Vleck [3o] was able to connect, in certain cases, continued fractions of the form (1.6) with definite integrals of the form f(t) ^t £-(z + t)^^' where f(t) > 0. However, H. Hamburger [6] was the first to develop a complete extension of Stieltjes' theory to continued fractions of the form (1.6) in 1920. In 1914 Hellinger and Toeplitz [7] established the fundamentals of a matrix theory for continued fractions of the form (1.6) with b. > 0 and a. real, i=l, 2, 3, ... In 1922 Hellinger [s] obtained a completion of this theory. Around 1900 Pringsheim [l9, 20] obtained answers to questions about convergence of continued fractions with complex elements. These continued fractions were of the form (1.7) ^ Zg 1 + ^ ^2 1 + 1 + '. Pringsheim found thkt continued fractions of the form (1.7) converge for Jz. ^ (1 - r._,)r., where I 0<r._, <1, i = l, 2, 3, ... Van Vleck [28, 29], about the same time, obtained similar results. Van Vleck [28, 29] also found that continued fractions of the form (1.8) ^1 •" 1 ^2 ^ ^TT-^ converge when z^ / 0 and |,^(z^)| ^ k/((z^) for k > 0 and i = 1, 2, 3, ... if and only if the series ^JI^J diverge; In 1912-1915 Szasz [26, 27j found that continued fractions of the form (1.7) converge if the series 2^|z.| converges and yjlzj^l ~ R^^±^J ^ 2. Extensions of the above mentioned results, as well as many new results, within a larger analytic theory may be found due to H. S. Wall and his associates. Specifically with respect to expression (1.7) and expression (1.8), Wall and his associates found that the inequalities of Pringsheim and Van Vleck restricting the z. to lie in a neighborhood of the origin could be replaced with inequalities restricting the z. to lie in a domain bounded by certain parabolas with foci at the origin (Scott and Wall [22J and Paydon and Wall [l8J ) . Wall and his associates also extended Stieltjes* theory to a class of continued fractions of the form (1.6) with a. and b. complex, i = 1, 2, 3, ... (Hellinger and Wall [9], Wall and Wetzel [si, 32], Dennis and Wall [3]). In 1948 Wall [34] published his Analytic Theory of Continued Fractions which serves as a unified theory for continued fractions and their application. It may be noted that this excellent book is widely considered to be the authoritative text on the analytic theory of continued fractions. CHAPTER II INTRODUCTORY CONCEPTS Definitions and Notation An expression of the form bg (2.1) a^ + ^ 0 b^ 2 a, + 1 b_. a^ + "^ 2 a-3 + •.• is called a continued fraction. In general, the elements a^, a, , a^j ... and b, , b^, b^, ... may be real or complex numbers, or they may be real or complex valued functions of one or more variables. The number of terms in a continued fraction may be finite or infinite. In view of the notational difficulties involved in expression (2.1), it is more commonly accepted to repre­ sent a continued fraction by an expression of the form a + ^ ^ ^ 0 a, + a^ + a.^ + • • • , or by an expression of the form ,00 w V 1 • The elements a and b^, for any integer n, are called the coefficients of the continued fraction (2.1) In particular, the elements b, , b^ > b.^, ••. are called the partial numerators, a-^ is called the leading constant and the elements a-,, a2» a^> ••' are called the partial b denominators. The fraction —, for any integer n, is an called the n partial quotient. The terminating continued fraction b, ^2-^3 b P 0 a, + a^ + a- + • • • + a Q 1 Z o n n is called the n convergent or n approximation of the continued fraction (2.1). We shall assume that all of the coefficients of a continued fraction are finite and that no partial denomi­ nator is ever zero. Successive Convergents We note from expression (2.2) that Pi ,^ «1 ' h ^^ ^ = 8 ar0^ + a^a2 + b2 _ ^0^1^2 •" ^0^2 "• ^2^1 The last of these relations indicates, in the general case, that for all integers n^ 2, P a P T + b P ^ / 9 ON _J}. _ n n-1 n n-2 n n^n-1 n^n-2 To prove relation (2.3) we use induction and note P P , that in order to progress from -^ to ^ we must replace , ^n n+1 a^ with a_ + n+1 . Then n n a n+J 1T b ^ a P T + " P T+bP^ r,^-U P .T n n-1 a ^, n-1 n n-2 a b ^TP , n+1 _ n+1 _ n+ ^^P1 n + n+1 n-1 °n.l \ , h^ Q " ^n.A ^ Vl°„-1- n n-1 aTV^n-l n^n-2 n+1 Thus equation (2.3) is valid for all integers n ^^ 2, and hence provides a means by which successive convergents of order n ^ 2 can be computed for the continued fraction (2.1). Further, we note from the above development that P =aP T+bP ^ and Q = a Q , + b^Q ^ for all n n n-1 n n-2 n n n-1 n n-2 integers n > 2. In order that equation (2.3) be valid for n = 1, we define P_, = 1 and Q_, = 0. We now consider the difference of two successive convergents of the continued fraction (2.1). That is, consider P P , P Q , - P ,Q (-y A\ _IL n-1 _ n n-1 n-1 n K^'^) Q " Q , " Q Q • n "^n-l ^n-l^n With the help of equation (2.3) we may substitute the appropriate values for P^ and Q in the numerator of the n n right-hand side of equation (2.4) to obtain ^ ^ILzl ^^n^n-1^ ^n^n-2^Qn-l " ^^n^n-l " ^nQn-2)Pn-l ^n " ^n-l " ^n-A ,.
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