Exponential expansion function

This property can be derived by means of a series expansion of the exponential function in eq. (39.43) and by neglecting higher order terms in 0. 

We will skip the algebraic details. The simple idea is that we can do long division of a function of the form in Eq. (3-30) and match the terms to a Taylor s expansion of the exponential function. If we do, we ll find that the (1/1) Pade approximation is equivalent to a third order Taylor... 

One idea (not that we really do that) is to apply the Taylor series expansion on the exponential function of A, and evaluate the state transition matrix with... 

GY noticed that the polarizability of an atom is approximately an exponential function of charge, and that the polarizability correction provided by the CPE expansion for an isolated atom was equal to the inverse of the Coulomb self energy of the... 

In general, the donor and receiver concentrations are exponential functions of time. It is only within the early time period when no more than 10-15% of CD(0) has been transported that the kinetics are essentially linear hence, the tmncated Maclaurin s expansion of Eqs. (5) and (7) leads to the linear relationships... 

Expansion of Pesc in powers of E is generally not recommended, because that expansion truncated to a finite number of terms results in large error as in a similar expansion of an exponential function of large argument. 

As a trivial example, think of the exponential function exp(—t) for positive times Its series expansion contains all positive powers of time, which grow indehnitely for a long time ... 

In a further simplification, namely the expansion of the exponential function in Eq. (2.13) into a Taylor series up to the linear term only (neglecting other terms which may be shown to be negligible) yields... 

This expansion may be compared with an expansion of the exponential function, exp icot, in the inverse Fourier transform of the spectral density,... 

There is also a relation between the amount of ionic character of a single bond and the enthalpy of formation of the bond. The amount of ionic character in percentage is roughly equal numerically to the heat of formation in kcaJ/mole. In applying this rule one must, of course, correct the heat of formation for the special stability of oxygen and nitrogen in their standard states, as expressed in Equation 3-13. This relation may be derived by expanding the exponential-function in Equation 3-15. The first term in the expansion, i(xA — zb)2, may be compared with the term 23 (xa — xb)2 of Equation 3-13. 

Of course, operating on the HF wave function with T is, in essence, full Cl (more accurately, in full Cl one applies 1 + T), so one may legitimately ask what advantage is afforded by the use of the exponential of T in Eq. (7.49). The answer lies in the consequences associated with truncation of T. For instance, let us say that we only want to consider the double excitation operator, i.e., we make the approximation T = T2. In that case, Taylor expansion of the exponential function in Eq. (7.49) gives... 

It should be stressed once more that the accumulation curve n(t) (or U(t)), especially at high doses, cannot be described by a simple equation (7.1.53) which is often used for interpreting the real experimental data (e.g., [19, 20]). Despite there is the only recombination mechanism, the A + B —> 0 accumulation kinetics at long t due to many-particle effects is no longer exponential function of time (dose). Therefore, successful expansion of the experimental accumulation curve U = U(t) in several exponentials (stages) does not mean that several different mechanisms of defect creation are necessarily involved (as sometimes they suggest, e.g., [39, 40]). 

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 ... 

Since the exponential function may be defined everywhere in the complex plane, we may expand exp(i0) and, using the series expansions for the trigonometric functions, obtain Euler s formula... 

It is noted that the right-hand side of Eq. (10.20) is just the series expansion of an exponential function. Therefore the overall residence time distribution probability density function in the SCISR is obtained to be... 

The time-local approach is based on the Hashitsume-Shibata-Takahashi identity and is also denoted as time-convolutionless formalism [43], partial time ordering prescription (POP) [40-42], or Tokuyama-Mori approach [46]. This can be derived formally from a second-order cumulant expansion of the time-ordered exponential function and yields a resummation of the COP expression [40,42]. Sometimes the approach is also called the time-dependent Redfield theory [47]. As was shown by Gzyl [48] the time-convolutionless formulation of Shibata et al. [10,11] is equivalent to the antecedent version by Fulinski and Kramarczyk [49, 50]. Using the Hashitsume-Shibata-Takahashi identity whose derivation is reviewed in the appendix, one yields in second-order in the system-bath coupling [51]... 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

This has an exponential of a matrix. It is defined in terms of the expansion of the exponential function, see for example Smith [514, p. 134-5]. Now, the usual eigenvalue-eigenvector equation can be written in compact form,... 

Pertsov et al. [19] and Kann [20] have proposed a logarithmic time dependence of the foam expansion ratio in order to determine the rate of bubble expansion. This approach is reasoned by the fact that at hydrostatic equilibrium further increase in foam expansion ratio occurs only when excess liquid is released with decreasing foam dispersity. It was experimentally established that at the final stage of internal foam collapse this increase in foam expansion ratio can be expressed by an exponential function [19,21]... 

The HF CO method is especially efficient if the Bloch orbitals are calculated in the form of a linear combination of atomic orbitals (LCAO)1 2 since in this case the large amount of experience collected in the field of molecular quantum mechanics can be used in crystal HF studies. The atomic basis orbitals applied for the above mentioned expansion are usually optimized in atoms and molecules. They can be Slater-type exponential functions if the integrals are evaluated in momentum space3 or Gaussian orbitals if one prefers to work in configuration space. The specific computational problems arising from the infinite periodic crystal potential will be discussed later. 

The same result follows when the average of the exponential function given in Eq. (51) is transformed using the cumulant expansion theorem and, assuming a Gaussian process, all correlations higher than second order (Stepisnik, 1981, 1985) are neglected. The particle velocity autocorrelations form a tensor... 

On the other hand, the perturbative approach for long-range interactions, that decompose the energy into terms of clear physical meaning is quite helpful in the development of a model. Hence, from the multipolar expansion of each term, one is able to know the form of the potential dependence on the distance. As to the short range part, the perturbative approach also indicates that the above long-range part must be supplemented by repulsive terms, that are well described by exponential functions [45,75]. 

The more recent ASP-W2 and ASP-W4 improve the accuracy of the description of multipoles, which is at multireference Cl level instead of MP2. All three potentials share the same form of repulsion and polarization term, the former using three sites with exponential functions while the latter is described by anisotropic dipolar and quadrupolar polarizabilities centered on the oxygen. Electrostatic interactions are modeled by a single-center (APS-W) or three-center (ASP-W2 and ASP-W4) multipolar expansion up to quadrupolar term for ASP-W and APS-W2 and up to hexadecapolar term for APS-W4. 

Higher order terms (e.g., Tl) do not appear, of course, because our example system contains only four electrons. If we remember that Ti and T2 commute, then all the terms from the equation above match those from the power series expansion of an exponential function Thus, the general expression for Eq. [30]... 

This result shows that for small n the number of trials required to find a step that reduces the value of the objective function, Tj(n T), grows only as the square root of n with a small coefficient. For moderately large n [provided n < (nil - it is necessary to retain the quadratic term in the expansion of the exponential function and T, (n T) becomes proportional to n with coefficient of proportionality less than one. This result was deduced by the authors in [5]. For values of (n-2) (— - greater than one the denominator becomes exponentially small and the approximation tc T(n T) begins to increase accordingly. 

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