Asymptotic approximation general examples

We showed in Section 2.3 that the real and imaginary parts of the electric susceptibility are connected by the dispersion relations (2.36) and (2.37). This followed as a consequence of the linear causal relation between the electric field and polarization together with the vanishing of x(<°) in the limit of infinite frequency to. We also stated that, in general, similar relations are expected to hold for any frequency-dependent function that connects an output with an input in a linear causal way. An example is the amplitude scattering matrix (4.75) the scattered field is linearly related to the incident field. Moreover, this relation must be causal the scattered field cannot precede in time the incident field that excited it. Therefore, the matrix elements should satisfy dispersion relations. In particular, this is true for the forward direction 6 = 0°. But 5(0°, to) does not have the required asymptotic behavior it is clear from the diffraction theory approximation (4.73) that for sufficiently large frequencies, 5(0°, to) is proportional to to2. Nevertheless, only minor fiddling with S makes it behave properly the function... 

This approximate method has been generalized to cases other than a spectral density on a finite interval. For example, if a spectral density is known to exponentially fall off at Targe to, then the continued fraction coefficients grow asymptotically as n2 at large n. Then if enough moments are known so that one approaches this asymptotic behavior, a similar estimate of the spectral density can be made. 

The present idea is to replace the ZORA ansatz, which already is an approximation to the energy-dependent elimination of the small component approach, with another but similar expression that relates the large and the small components. The general ansatz function should have the same shape as the ZORA function close to the nucleus. Its first derivative should also be reminiscent of that of the ZORA function. A general function f[r) that fulfills the desired asymptotic conditions for r —0 and for r -> < can for example consist of one exponential function or of a linear combination of a couple of exponential functions as... 

The consideration that the velocity of electrons is much higher than that of the nuclei (a consequence of their much smaller masses) leads to the Born-Oppenheimer approximation, perhaps the better known example of the near separability of variables. We reconsider it in view of subsequent generalizations. Using the language of classical mechanics, we will speak of adiabatic separability, which can be shown to be related to a semiclassical expansion, i.e. to an asymptotic expansion in h. See references [11-14] where we also discuss a post-adiabatic representation. 

In the experiment one often deals with large initial deviations from equilibrium for example, such is the case when a new oil-water interface is formed by the breaking of larger emulsion drops during emulsification. In the case of large perturbation there is no general analytical expression for the dynamic surface tension a(t) since the adsorption isotherms (except that of Henry, see Table 1) are nonlinear. In this case one can use eiflier a computer solution (44, 45) or apply the von Karman approximate approach (46,47). Analytical asymptotic expressions for the long time (t tj) relaxation of surface tension of a nonionic surfactant solution was obtained by Hansen (48) ... 

In case the viscoelastic liquid is only slightly elastic, the general constitutive behavior depicted by Eq. 1 can be substantially simplified, by obtaining an asymptotic series of approximation in W, which can be taken as small under these conditions. As a consequence, one may arrive at the following second-order constitutive model, for example, [2] ... 

The sum (6.50) can be calculated for k kj, for example, by the Ewald method. However, for k = kj the series (6.50) appears to be divergent [95]. This divergence is the result of the general asymptotic properties of the approximate density matrix calculated by the summation over the special poits of BZ (see Sect. 4.3.3). The difficulties connected with the divergence of lattice sums in the exchange part have been resolved in CNDO calculations of solids by introduction of an interaction radius... 

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