Exponential power series

An exponential power series is recuperated by dividing through by the first term, Eq. (A2.91). Where the total mean square displacement tensor y4 is a spheroid that deviates little from the isotropic, i.e. one that is almost a sphere, then terms linear in A will be adequate [6]. This is the almost isotropic approximation and Eq. (A2.87) becomes Eq. (2,41) for = 1. 

A numerically specified function, that is a dataset, can almost always be made to fit one or more of the following sum of exponentials, power series or Fourier series regardless of its physical significance. Acton s (1970) chapter entitled What not to compute is essential reading. Ashe points out most of us can more easily compute than think . None the less, if correctly used, digital collection and computer analysis of data have brought many insoluble problems into the realm of the soluble. 

The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... 

It we now expand the exponential in a power series, we obtain the following expansion which places the various joint moments of fa and fa in evidence... 

Expanding the second exponential in the right-hand side of Eq. (3-266) in a power series and making use of Eq. (3-261), we obtain... 

Equation (7.25) can be substituted into equation (7.20) to give a second order differential equation in ijj. In theory, the resulting equation can be solved to give ip as a function of r. However, it has an exponential term in -ip, that makes it impossible to solve analytically. In the Debye-Hiickel approximation, the exponential is expanded in a power series to give... 

The variable s is a dummy variable in the sense that it does not enter die final result. Thus, if the exponential function in Eq. (94) is expanded in a power series in s, the coefficients of successive powers of s are just the Hermite polynomials divided by u . It is not too difficult to show that Eqs. (93) and (94) are equivalent definitions of the Hermite polynomials. 

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting... 

A well behaved function is one that grows slowly enough at infinity so that the integrand is null. Since the PDF typically falls off exponentially at infinity, any function that can be expressed as a convergent power series will be well behaved. 

This is an analogue of the classical moments-generating functional discussed by Kubo [39]. Upon expanding the exponential as a power series, the operator J f acts to place each term in so-called normal order, in which all creation operators are to the left of all annihilahon operators j/. By virtue of this ordering (and only by virtue of this ordering). 

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... 

An inspection of the power series expansion for the exponential in Eqs. (15.40) and (15.41) indicates that neither expression diverges as a and become arbitrarily close to equal (an analogous consideration of the power series expansion for the sine function in Eq. (15.40) indicates the first term on the r.h.s. to be similarly free from singularities). 

We introduce the first of the Debye-Hiickel approximations by considering only those situations for which < kBT). In this case the exponentials in Equation (28) may be expanded (see Appendix A) as a power series. If only first-order terms in z,eyp/kBT) are... 

The constants D and a were evaluated from two sets of the experimental release data using Equation 8. Although Equation 8 is in the form of an infinite series, numerical evaluation readily shows that the first term predominates even for very short times. This is also obvious from the plot in Figure 5 where it can be seen that a straight line may be drawn through the points starting from t = 0. Thus, the slopes of the lines taken after a reasonably short time will yield the exponential power, —fi2Dt/l2, where f3 is the first positive root of Equation 9. 

To obtain a useful approximate solution of the PB equation (S8.6-4), we consider the dilute limit in which the electric potential O is weak compared with the ambient thermal energy kT. In this limit, the Boltzmann exponential can be linearized by retaining only the leading term in the power series expansion... 

In order to estimate the transcendental number e, we will expand the exponential function ex in a power series using a simple iterative procedure starting from its definition Eq. (25) together with Eq. (12). As a prelude, we first find the power series expansion of the geometric series y — 1/(1 + x), iterating the equivalent expression ... 

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. 

A quantum-classical approximation for eq.(5) can be obtained by expanding the exponential operator in a power series of the reduced Planck constant h (or, equivalently, in a power series of (m/M)1/2 [11])... 

Equations (4.212) are solved sequentially beginning from Eq. (4.213) by expanding in a power series with respect tox. The right-hand sides of Eqs. (4.212) and (4.213) are proportional to an exponentially small parameter A,. For this reason alone, we did not take into account the corrections of the order ul n l when deriving Eqs. (4.167). However, the quantities... 

Since at microwave frequencies and normal temperatures, hv kT, we can expand the exponential as a convergent power series to yield... 

From these powers, one can also construct power series, including special functions such as the exponential function... 

Simpler models for byproduct formation were developed from the same set of experiments. These models are of simple power-series or exponential form and serve adequately to predict the small amounts of byproducts formed by the reaction. 

In Eq. 11.16, we encountered a familiar operator 96, but in an exponential form. Many operators R can most conveniently be expressed in an exponential form, where the exponential is defined in terms of an infinite power series ... 

In order to use this equation, we are forced to make the approximation that zfii/iIDkT 1 so that the exponential can be expanded in a power series ... 

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