Asymptotic series

In certain circumstances, a divergent series can be used to determine approximate values of a function as x oo. Consider, as an example, the 

This is an instance of an asymptotic series, as indicated by the equivalence symbol rather than an equal sign. The series in brackets is actually divergent. However, a finite number of terms gives an approximation to erfc x for large values of x. The omitted terms, when expressed in their original form as an integral, as in Eqs. (7.118) or (7.119), approach zero as x oo. 

A convergent series, such as the ones considered earlier, approaches fix) for a given x as n oo, where n is the number of terms in the partial sum S . By contrast, an asymptotic series approaches fix) as x oo for a given n. 

The exponential integral is another function defined as a definite integral that cannot be evaluated in a simple closed form. In the usual notation. 

By repeated integration by parts, the following asymptotic series for the exponential integral can be derived  

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Eqs. (40)-(45) describe the behavior of a mass transport process occurring at very large Schmidt numbers. For finite values of Sc, Chin [16] extended the 0-series expansion analysis and obtained the following asymptotic series for Sfiav on a rotating hemisphere ... 

It has been generally assumed until recently that the series (6.1) is convergent, at least for sufficiently small z, though it was recognised that convergence was slow except for very small z (32). It has recently been shown that this series is divergent for any non-zero value of z (205) it is suggested that it is nevertheless useful as an asymptotic series, so that the coefficients are of interest. Values have been obtained for the first three coefficients for linear polymers. 

Another common example arises in the expansion of the interaction energy of two atoms as a function of the inverse interatomic distance 1/Rab- This turns out to be a Taylor series which differ from the correct energy by a real contribution with zero Taylor expansion. The difference may be attributed to so-called exchange repulsion. In this case, it is nowadays known that die Taylor series has a zero convergence radius, so that the energy expression constitutes an example of an asymptotic series (to be defined in a moment) which is non-convergent for all Rab-... 

Sometimes, one is not so much concerned with the pointwise convergence of a series one merely wants each partial sum to be asymptotically better than the last Such an asymptotic series is almost always an inverse power series, and it is then defined as follows ... 

A famous example of a non-convergent asymptotic series is the Stirling formula for the factorial function ... 

Note, however, that the common belief, that an asymptotic series is always non-convergent, is wrong. 

Ochkur128 has expanded the exchange amplitude in an asymptotic series in powers of 1 /k0 and has found its leading terms. He has found that the term with the operator l/r2 can be safely neglected, while the leading part of the term describing the interelectronic interaction can be found via the following substitution... 

This section is based mostly on the results presented in Ref. 78 and is arranged in the following way. In Section III.B.l we note mentioned the problem of superparamagnetic relaxation, which has been already tackled by means of the Kramers method, in the in Section II.A), and show how to obtain the analytical solution (in the form of asymptotic series) for the micromagnetic Fokker-Planck equation in the uniaxial case. In Section III.B.l the perturbative... 

It is noteworthy that Brown himself resumed [88,89] the studies on A-i and modified the preexponential factor in Eq. (4.132), transforming it into an asymptotic series in ct 1. On the basis of Eq. (4.128), he had constructed an integral recurrence procedure, and evaluated Xj down to terms oc 1/ct10. What we do below, is, in fact, carry on this line of analysis that had not been touched since then. Our method advances Brown s results in two aspects (1) for it is simpler, and (2) it provides not only the eigenvalue but the eigenfunction as well. Possessing only the latter, one is able to obtain theoretical expressions for the directly measurable quantities, that is, the susceptibilities yjk>. 

Note the appearance of the AB equilibrium geometry (Rm, which can be obtained by a self-consistent procedure136) and Le Roy s parameter142 [R0, which represents the smallest value of the internuclear distance for which the asymptotic series of the dispersion energy is still a good representation of the damped series (49)] in the definition of the reduced coordinate x represents the expectation value of the square of the radial coordinate for the outermost valence electrons, which is tabulated in the literature143 for atoms with 1 120. Other important parameters in the dispersion damping... 

For large positive values of jc, r(x) approximates the asymptotic series ... 

For real molecules A and B the charge clouds have exponential tails so that there is always some overlap and the expansion (16) is an asymptotic serie Still, for the long range the multipole approximation to can be quite... 

The pertinent indefinite integrals and approximating series were listed in Sect. 2.1.6. In practice, the value of the definite integral for 7s is usually much smaller than that for l and can be neglected. Taking only two terms in the asymptotic series, we obtain for the case of a constant temperature gradient that ... 

Assuming that the values of the integral at 7s can be abandoned compared with those for 7a, and taking into account only the first member of the asymptotic series, one comes to ... 

For real molecules A and B the charge clouds have exponential tails so that there is always some overlap and the expansion (16) is an asymptotic series . Still, for the long range the multipole approximation to AE can be quite accurate, if properly truncated (for instance, after the smallest term). For shorter distances, the penetration between the molecular charge clouds becomes significant, the screening of the nuclei by the electrons becomes incomplete even for neutral molecules, and the power law for AE 5 is modified by contributions which increase exponentially with decreasing R. These penetration contributions we define as ... 

Of course, the solution (4-181) is only the first approximation in the asymptotic series (4 175). In writing (4-177), we neglected certain smaller terms in the nondimensionalized equation, (4-170), because they were small compared with the terms that we kept. To obtain the governing equation for the second term in the boundary-layer region, we formally substitute the expansion, (4-175), into the governing equation, (4-170) ... 

Just as in mechanics of viscous fluids, the approximate solution of convective problems of mass and heat transfer is based on the methods of perturbation theory [96, 224, 258, 485], In these methods, the dimensionless Peclet number Pe occurring in Eq. (3.1.8) is assumed to be a small (or large) parameter, with respect to which one seeks the solutions in the form of asymptotic series. 

The asymptotic series provided by perturbation methods are of restricted scope. Moreover, one can usually obtain no more than two or three initial terms of the corresponding expansions. Thus, one cannot estimate the solution behavior for intermediate (finite) parameter values and there are severe limitations on the applicability of asymptotic formulas in the engineering practice. This is the most essential drawback of perturbation methods. 

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