On the Computation of Exponential and Hyperbolic Functions Using Continued Fractions

where |y1P #1. These were first discussed by Schur [6]1 with |7pl <1 in connection with functions bounded in the unit circle. In ?2 it is shown that to an arbitrary power series there correspond continued fractions (1.1), and, conversely, to every such continued fraction there corresponds a power series which agrees with the power series for the 2pth and (2p+1)th approximants of the continued fraction up to and including the terms involving zP and zP-1, respectively. An important new feature which is given in ?3 is the simple algorithm for the expansion of a power series into a continued fraction (1.1), and, conversely, for the computation of the corresponding power series from the continued fraction. Heretofore this computation has been exceedingly cumbersome. In ?4 conditions under which these continued fractions converge to the corresponding power series are considered, in particular, if the generating function is meromorphic. In ?5 convergence regions for the expansions (1.1) are found, and in ?6 convergence regions for the continued fraction whose approximants are the even and odd approximants of (1.1). In ?7 certain transformations of (1.1) are discussed.

A note on extending Euler's connection between continued fractions and power series

Hankel continued fraction and its applications

The prime goal of this paper is to establish sharp lower and upper bounds for useful functions such as the exponential functions, with a focus on exp(−x^2), the trigonometric functions (cosine and sine) and the hyperbolic functions (cosine and sine). The bounds obtained for hyperbolic cosine are very sharp. New proofs, refinements as well as new results are offered. Some graphical and numerical results illustrate the findings.

Meander permutations have been encountered in the context of Gauss words, singularity theory, Sturm global attractors, plane Cartesian billiards, and Temperley–Lieb algebras, among others. In this spirit, we attempt to investigate the difference of orderings of homoclinic orbits on the stable and unstable manifolds of a planar saddle. As an example, we consider reversible linear Anosov maps on the 2-torus, and their relation to continued fraction expansions.

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

converge to P(z) uniformly over an arbitrary closed region, K, exterior to the real axis. (ii) If P(z) is a Stieltjes power series with radius of convergence R>0, then an arbitrary infinite sequence of distinct approximants converges to P(z) uniformly over the circle Iz -=R-6, a > 0. (iii) If P(z) is a positive definite power series which is summable (Borel) to F(z), then Sk(k > 1) converges to F(z) uniformly over K. If the reciprocal of P(z) is summable (Borel) the same holds for k < 1. (iv) If P(z) is a Stieltjes series possessing minimal extensionsi D=-n Cif(-z)i for every n, and if the series EZiXo(1)ic-ilz-i-' is summable (Borel) to F(z), then in the Pade table for P(z), S-1 converges to F(z) uniformly over an arbitrary closed region exterior to the negative half of the real axis. As a corollary to (iii) we find that when P(z) is summable (Borel) and has a corresponding continued fraction, the latter is necessarily convergent to the Borel sum. The converse is not true. That is, the corresponding continued fraction may converge when P(z) is not summable (Borel). 2. The convergence of the diagonal files in the Pade table for a convergent positive definite power series. We recall that P(z) is a positive definite series if all the quadratic forms j= n= , 1, 2, * , are positive de-

Flat Power Series over a Finite Field

Constitutive behavior and processing maps of low-expansion GH909 superalloy

In this paper, two design procedures for sigmoid function using exponential function have been proposed implementing division operation and feedback methodology. Hyperbolic tangent function has also been synthesized using the later. All the proposed circuits along with their building blocks have been implemented using current mode device Operational Transconductance Amplifier (OTA). Sensitivities of the circuit and process parameters on the output have been calculated. Performances of all synthesized circuits have been verified with SPICE simulation, establishing the effectiveness of the proposed methodology.

In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose being to point out a common origin with another analogue family in odd characteristic, which has been studied in [LY1].

A numerical method for high order approximation of u(t) = \exp (tA)_{u_0} , where A is an N × N matrix and u_0 is an N dimensional vector, based on the continued fraction expansion of \exp z is given. The approximants H_k(z) of the continued fraction expansion of \exp z are shown to satisfy |H_k(z)| ≤ 1 for \mathrm{Re} z ≤ 0 , which results in an unconditionally stable method when every eigenvalue of A lies in the left half-plane or on the imaginary axis.

CHAPTER XII - INVERSE CIRCULAR FUNCTIONS, HYPERBOLIC FUNCTIONS AND INVERSE HYPERBOLIC FUNCTIONS

Theoretical results are derived for constructing continued fractions which correspond, in some prescribed way, to given power series expansions at two points. In those cases where the two power series are expansions of the same function, new approximations to the functions are obtained. Error estimates for such rational function approximations are developed in certain cases. Some examples are given.

On lacunary formal power series and their continued fraction expansion