Rational function

Partial Fractions Rational functions are of the type f x)/g x) where /x) and g(x) are polynomial expressions of degrees m and n respectively. If the degree of/is higher than g, perform the algebraic division—the remainder will then be at least one degree less than the denominator. Consider the following types ... 

The Rational Function Optimization (RFO) expands the function in terms of a rational approximation instead of a straight second-order Taylor series (eq. (14.3)). 

In the Partitioned Rational Function Optimization (P-RFO), two shift parameters are employed. 

Functions with localized strong inflections or poles may be approximated by rational functions of the general form... 

Since the machine performs only arithmetic operations (and these only approximately), iff is anything but a rational function it must be approximated by a rational function, e.g., by a finite number of terms in a Taylor expansion. If this rational approximation is denoted by fat this gives rise to an error fix ) — fa(x ), generally called the truncation error. Finally, since even the arithmetic operations are carried out only approximately in the machine, not even fjx ) can usually be found exactly, and still a third type of error results, fa(x ) — / ( ) called generated error, where / ( ) is the number actually produced by the machine. Thus, the total error is the sum of these... 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

The plasma concentration function Cp in the time domain is obtained by applying the inverse Laplace transform to the two rational functions in the expression for Cp in eq. (39.58) ... 

The second class of GGA exchange functionals use for F a rational function of the reduced density gradient. Prominent representatives are the early functionals by Becke, 1986 (B86) and Perdew, 1986 (P), the functional by Lacks and Gordon, 1993 (LG) or the recent implementation of Perdew, Burke, and Emzerhof, 1996 (PBE). As an example, we explicitly write down F of Perdew s 1986 exchange functional, which, just as for the more recent PBE functional, is free of semiempirical parameters ... 

Indefinite Integration. KACSYMA can handle integrals involving rational functions and combinations of rational, algebraic functions, and the elementary transcendental functions. It also has knowledge about error functions and some of the higher transcendental functions. 

As an alternative procedure to predict coefficients of a radial function p(x) for electric dipolar moment, one might attempt to convert the latter function from polynomial form, as in formula 91, which has unreliable properties beyond its range of validity from experimental data, into a rational function [13] that conforms to properties of electric dipolar moment as a function of intemuclear distance R towards limits of united and separate atoms. When such a rational function is constrained to yield the values of its derivatives the same as coefficients pj in a polynomial representation, that rational function becomes a Fade approximant. For CO an appropriate formula that conforms to properties described above would be... 

Although transformation of coefficients pj into coefficients qj is readily practicable, the resulting values for CO adopt unwieldy magnitudes. Chackerian and Tipping [141] fitted a function of the latter form from experimental and theoretical (computations of molecular electronic structure) information in judicious combination, according to which they calculated vibration-rotational matrix elements for transitions in bands 5-0 and 6-0 fitting the latter values with formula 84 yielded the values of quantities presented above. Rational functions, such as those in formulae 92 - 94 or others, transcend the spirit of Dunham s approach because their construction incorporates physical knowledge of a quantity that is superfluous for invocation of a mere truncated polynomial. 

Selected entries from Methods in Enzymology [vol, page(s)] Graphical techniques, 210, 306 polynomial methods, 210, 307 with rational functions, 210, 311 with spline functions, 210, 312 with trigonometric functions, 210, 312... 

We note the possible similarities of the effective model (42) with Golay s theory as presented in Paine et al. (1983). In the effective dispersion term this theory predicts a rational function of and we confirm it. Nevertheless, there is a difference in particular coefficients. 

In our assumptions, system (27) has the finite number of roots (by Lemma 14.2 in Bykov et al., 1998), so that the product in Equation (26) is well defined. We can interpret formula (26) as a corollary of Poisson formula for the classic resultant of homogeneous system of forms (i.e. the Macaulay (or Classic) resultant, see Gel fand et al., 1994). Moreover, the product Res(R) in Equation (26) is a polynomial of R-variable and it is a rational function of kinetic parameters fg and Tg (see a book by Bykov et al., 1998, Chapter 14). It is the same as the classic resultant (which is an irreducible polynomial (Macaulay, 1916 van der Waerden, 1971) up to constant in R multiplier. In many cases, finding resultant allows to solve the system (21) for all variables. ... 

The four-term overall reaction rate equation. It follows from Propositions 1, 3 and the fact that the kinetic polynomial defined by formula (26) is a rational function of reaction weights fs and that we can write Equation (67) as... 

The differential equation (2.48) is separable, and by integrating the rational function on its right-hand side the solution is given by... 

Rational functions are frequently encountered as rate expressions of catalytic reactions. In addition, the function... 

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