Poles second order

The first major pole is contributed by the output L-C filter. It represents a second order pole which exhibits a Q phenomenon, which is typically ignored, and a -40dB/decade rolloff above its corner frequency. The phase plot will quickly begin to lag starting at a frequency of 1/lOth the corner frequency, and will reach the full 180 degrees of lag at 10 times the corner frequency. The location of this double pole is found from... 

The Second Order Pole/Zero (BiQud) HIter... 

First-Order Interpolation. The dimension dependence is dominated by singularities at i = 1, arising from a second-order pole (like the hydrogenic atom but with a different residue) and a confluent first-order pole. Deducting the readily calculable contributions from these poles markedly improves the efficacy of dimensional interpolation. The simplest approximation of this kind yields... 

Figure 13. Cumulative contributions (% of total) to electronic energy of ground-state H2 arising from dimensional singularities. For scaled distance Ru = 1. Lowest curve (dashed) shows contribution from second-order pole at U = 1 middle curve (dot-dashed) sum of first- and second-order poles at > = 1 uppermost curve (solid) adds contributions from singularties at 00 limit.

With appropriate dimensional scalings, theD — 1 limit of Schrodinger equations for coulombic systems provides important information about the dimension dependence of energy eigenvalues. The energy typically has a second-order pole at D = 1, the residue of which can often be exactly determined. We demonstrate this with some simple examples and then review a systematic procedure for characterizing a class of dimensional singularities found in coulombic problems. 

A characteristic feature of ground state energies of coulombic systems is the presence of a second-order pole at = 1. The origin of these poles has been explained by Doren and Herschbach [14,16] in terms of an analysis of the Schrodinger equation at particle coalescences. Their method of analysis allows one to predict the locations and types of a certain class of dimensional singularities without actually having to solve for the function E S). We will illustrate this first for central-potential problems and then for many-particle systems. 

This singularity analysis is applicable to any Schrodinger equation for a central potential, and the resulting singularity structure depends only on the behavior of the potential in the limit r — 0. Any system whose potentied at small r is proportional r will have a second-order pole according to Eq. (30). However, it is important to note that this second-order pole is only the most singular behavior at i = 3 — 2n in general, one can also expect a coincident first-order pole. 

In principle there could also be other types of singularities at D = 3—2n as long as they do not diverge as quickly as a second-order pole. In the case of the one-electron atom the residue a i of the first-order... 

A characteristic feature of E 6) for systems with coulomb potentials is the presence of a second-order pole at = 1. Indeed, the exact solution for the ground state of the one-electron atom is simply [25]... 

It is clear that the only kinds of singularities present in Eq. (15) are poles, corresponding to the zeros of the denominator polynomial, and we find that most of the Fade approximants for the S expansions of hydrogen, helium, and have a pair of poles very close to 6 = 1, which models the second-order pole in Eqs. (10) and (12). If we subtract from the power series the 6 expansion of, with a 2... 

According to Eq. (12), the coulombic singularities have the form of a second-order pole and a confluent first-order pole. The residues of the poles can in principle be calculated exactly, from the D 1 limit of the Schrodinger equation, as we discussed in Chapter 4.1. This suggests that we use an approximant of the form... 

For n = 0 the shifted expansion is identical to the rescaled expansion, Eq. (41), with a second-order pole at = 1, but they differ at higher order since the shifted expansion adds additional higher-order poles at <5 = 1. We know from dimensional singularity analysis [18] that there are no poles at = 1 of order higher than two, so we can expect that the shifted expansion will be less and less accurate than the rescaled expansion as n increases. This is indeed what happens in practice [11]. The shifted expansion procedure has been rather popular [31], but now that we have information about the dimensional singularity structure of the problem this method should probably be considered obsolete. 

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