Truncated power series

The simplest form of approximation to a continuous function is some polynomial. Continuous functions may be approximated in order to provide a simpler form than the original function. Truncated power series representations (such as the Taylor series) are one class of polynomial approximations. 

In thip appendix, a summary of the error propagation equations and objective functions used for standard characterization techniques are presented. These equations are Important for the evaluation of the errors associated with static measurements on the whole polymers and for the subsequent statistical comparison with the SEC estimates (see references 26 and 2J for a more detailed discussion of the equations). Among the models most widely used to correlate measured variables and polymer properties is the truncated power series model... 

In Eq. (12.16), one may imagine taking X intervals so small that AE on any given interval is arbitrarily close to zero. In that case, we may represent the exponential as a truncated power series, deriving... 

In the derivation of the simplified expressions for solubility and diffusion coefficients, eqs. (4) and (9), C was assumed to be small. This fact does not limit the usefulness of these expressions for high concentrations. We show below that sorption and transport expressions, eqs. (11) and (14), respectively, derived from the simplified equations retain the proper functional form for describing experimental data without being needlessly cumbersome. Of course, the values of the parameters in eqs. (4) and (9) will differ from the corresponding parameters in eqs. (3) and (8), to compensate for the fact that the truncated power series used in eqs. (4) and (9) poorly represent the exponentials when aC>l or 0C>1. Nevertheless, this does not hinder the use of the simplified equations for making correlation between gas-polymer systems. 

The matrix [3] is then obtained by inversion of the result of this series. The series representation (Eq. A.6.6) is preferred to Sylvester s formula (especially when the order of the matrix is > 3 or 4) but is not as fast as the truncated power series (Eq. A.6.4) (Taylor and Webb, 1981). For problems involving a singular, or nearly singular [C>], the series (Eq. A.6.7) is the best alternative to Sylvester s formula. 

The vibrational energy is described empirically by a truncated power series in v + 1/2,... 

A strong numerical evidence supporting the above statements has been provided in Ref. [98]. Before discussing a few representative examples, let us emphasize that in all numerical calculations reported in Ref. [98] and reviewed here, we tested, for the first time ever, the exact theory, in which we used the unexpanded form of the exponential operator to define the E X) energy rather than the truncated power series expansion in X used in Ref. [97]. This was made possible by representing the operators H and X as matrices in the finite-dimensional 7V-electron Hilbert spaces relevant to a molecular system under consideration (using all symmetry-adapted Slater determinants ) and n = 1,..., TV, defining the... 

This can be achieved most easily with a careful choice of internal coordinates. Since anharmonic force fields are usually expanded only up to fourth order, it is of considerable importance that cubic and quartic interaction terms in internal coordinates are minimized due to the curvilinear nature of these coordinates. However, this does not mean that a truncated power series expansion will behave correctly farther away from the reference geometry usual expansions suffer from convergence problems. 

Appendix A includes a sechon on curve fitting that outlines a means to find the cs for a truncated power series expansion. We have, then, a general approach for finding an expression for X in ferms of P, V, and T. When that is obtained, we return to Equation 2.28 and write X = PV - nRT, now using the fitting expression in P, V, and T in place of X. Such a result is a real gas law based on the PVT laboratory data that are available. 

From statistical mechanics, discussed later in this text, one can directly obtain heat capacities of certain substances from information about the quantum mechanical energy levels of the gas particles. However, for temperature ranges from about 300 K to at least 1000 K higher, direct calorimetric measurements have shown very slight variation in the heat capacities of monatomic gases with temperature. Molecular gases tend to show a dependence on temperature that can often be well represented by a truncated power series expansion (see Appendix A) of the heat capacity per mole ... 

Expressing a function as a polynomial in the variable(s) of the function constitutes a power series form of the function. Power series expansions are not usually applied to simple polynomials, for they already are power series. A power series expansion of sin(x), or exp(x), however, is a way of expressing those functions in terms of a pol)momial in x. The order of the expansion determines the accuracy, with an infinite expansion being precisely equivalent to the expanded function. Sometimes, the function that we seek to expand is unknown, but its low-order derivatives may be known. In that situation, a truncated power series expansion provides an approximate form of the unknown function one can find an approximation without having to know the function ... 

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