Rational Approximations for Irrational Functions using Collocating Continued Fraction Expansions

<div> <div> <div> <p>This paper is motivated by the need in certain engineering contexts to construct approximations for irrational functions in the complex variable z. The main mathematical tool that will be used is a special continued fraction expansion that cause the rational approximant to collocate the irrational function at specific important values of z. This paper introduces two theorems that facilitate the construction of the rational approxima- tion. </p> </div> </div> </div>

Rational Approximations for Irrational Functions using Collocating Continued Fraction Expansions

<div> <div> <div> <p>This paper is motivated by the need in certain engineering contexts to construct approximations for irrational functions in the complex variable z. The main mathematical tool that will be used is a special continued fraction expansion that cause the rational approximant to collocate the irrational function at specific important values of z. This paper introduces two theorems that facilitate the construction of the rational approxima- tion. </p> </div> </div> </div>

The Imagery Models of Mathematics Teacher Candidates on Graph of Rational and Irrational Functions

Mathematics teacher candidates are required to have both expertise and communication skills. Therefore, they should have a good imagery model. The present study is a descriptive qualitative research that involved the students of mathematics education program, FKIP, Universitas PGRI Madiun. The result shows there are 10 models of rational-irrational function graphs that are proposed by the candidates of mathematics teacher involved in the present study.. Mathematics lectures should enrich learning activities and materials to stimulate and develop students' capability in devising the graphs of rational-irrational functions.

Approximation of Irrational Matrix Functions and Its Application to Continuous-Discrete Model Conversion

In this paper, an improved method of rational approximation is presented for evaluating the irrational matrix function f(A), where A is a square matrix and f(s) is a scalar irrational function which is analytic on the spectrum of A. The improvement in the accuracy of the approximation off (A) by a rational matrix function is achieved by using the multipoint Pade approximants to f(s). An application example to model conversion involving the evaluations of the matrix exponential exp (AT) and the matrix logarithm ln(F) is provided to illustrate the superiority of the method.

XXI. A Memoir on the Single and Double Theta-Functions

The Theta-Functions, although arising historically from the Elliptic Functions, may be considered as in order of simplicity preceding these, and connecting themselves directly with the exponential function (e x or) exp. x ; viz., they may be defined each of them as a sum of a series of exponentials, singly infinite in the case of the single functions, doubly infinite in the case of the double functions ; and so on. The number of the single functions is = 4; and the quotients of these, or say three of them each divided by the fourth, are the elliptic functions sn, cn, d n ; the number of the double functions is (4 2 = ) 16 ; and the quotients of these, or say fifteen of them each divided by the sixteenth, are the hyper-elliptic functions of two arguments depending on the square root of a sex tic function : generally the number of the p -tuple theta-functions is = 4 p ; and the quotients of these, or say all but one of them each divided by the remaining function, are the Abelian functions of p arguments depending on the irrational function y defined by the equation F ( x, y ) = 0 of a curve of deficiency p ). If instead of connecting the ratios of the functions with a plane curve we consider the functions themselves as coordinates of a point in a (4 p —1)dimensional space, then we have the single functions as the four coordinates of a point on a quadri-quadric curve (one-fold locus) in ordinary space; and the double functions as the sixteen coordinates of a point on a quadri-quadric two-fold locus in 15-dimensional space, the deficiency of this two-fold locus being of course = 2.