Laurent Expansions

The reducible Green function is then given by the coefficient Co in the Laurent expansion of the full Green function... 

The Laurent expansion is very useful in analyzing the nature of singularities. However, a discussion of this aspect would go way beyond the scope of this Appendix. Therefore, we emphasize only the relation of the Laurent series... 

In 1973 Andersen and Woolley [1.25] extended the LCMTO method to molecular calculations. At the end of their paper they introduced that choice of MTO tail, i.e. proportional to J = 9i/j/9E, which in a natural fashion ensured orthogonality to the core states and at the same time led to an accurate and elegant formulation of linear methods. The resulting, technique was immediately developed in a paper by Andersen [1.26] which, in a condensed form, contains most of what one need know about the simple concepts of linear band theory. Thus, we find here the KKR equation within the atomic-sphere approximation at this stage is called ASM the LCMTO secular matrix, latter called the LMTO matrix the energy-independent structure constants and the canonical bands and the Laurent expansion of the logarithmic-derivative function and the corresponding potential parameters. 

Fig.3.1. Estimates of the logarithmic-derivative function for free s electrons compared with the exact result D(x) = x cot x - 1, x = S/E explained in Sect. 4.4. The curve labelled to is the second-order estimate (3.51), E(D) is the third-order estimate (3.50). while Lau is the Laurent expansion (3.30) valid to third order in (E - EV)S2. The potential parameters used in the three estimates are derived in Sect.4.4 and listed in Table 4.4. The two open circles in the figure refer to the points (EVS2,DV) and (EVS2,D ), where EVS2 is K2S2 of Table 4.4...

The results of this and Sect.3.3 are collated in Fig.3.1 and compared to an exact calculation of the logarithmic derivative function for free s electrons. The most striking features of this comparison are the very accurate description provided by both the Laurent expansion (3.30) and the variational estimate (3.50) in the region of negative D, and the quite erroneous estimate given by the variational expression in the region around D. ... 

The first term on the right-hand side may be found from (4.41), the second term from the Laurent expansion (3.32), i.e. 

The simplest way of evaluating the integral is to distort the path of integration and separate it into individual contours, each of which encloses one pole of the function. With the direction of rotation indicated in fig. 42, J is then equal to the negative sum of the residues of the integrand in these poles (the residue is 2m times the coefficient of 1 /(x—a) in the Laurent expansion in the neighbourhood of the pole a wo will use the symbol Res0 for the residue at the pole a) ... 

FIGURE 13.5 Contour for derivation of Laurent expansion of f z) about singular point z = Zq. The singularities at zi and Z2 are avoided. 

This function is regular at = 1, so we can expect that it will be more accurately modeled by partial sums than will the original function E. This summation technique was suggested by Doren and Herschbach [43], who called it the hybrid expansion, since it can be thought of as the sum of a truncated Laurent expansion about = 1 and a truncated Taylor expansion about = 0. 

This series is often called the Laurent expansion. If this were multiplied by it is clear that g(s) would in fact be an ascending series in s. The actual series for f(s) need not be known, but the above clearly represents its behavior if it is known that an Vth order pole exists. 

Hence, the value of the contour integral of f(s) is simply 2iri times the coefficient of in the Laurent expansion. The coefficient is called the residue of the function. 

The Laurent expansion is often not obvious in many practical applications, so additional procedures are needed. Often, the complex function appears as a ratio of polynomials... 

Now, if a simple pole exists at s = a, then obviously (s - a) must be a factor in gis), so we could express the denominator as g(s) = (s - a)G(s), provided G(s) contains no other singularities at s = a. Clearly, the Laurent expansion must exist, even though it may not be immediately apparent, and so we can always write a hypothetical representation of the type given in Eq. 9.75 ... 

Finally, the existence of x( ) fhe causality Fourier transform [Eq. (E.lb)] follows immediately that x z = x + iy) is an analytical function in the upper plane including the real axis. Moreover, x z) — 0 when y > 0 and 1 1 —> OO. These two properties allow Eq. (E.3) being readily evaluated via the contour integration formulation. Note also the Laurent expansion of... 

The eigenvalue equation (4.107) can be solved in several different ways. For example, it is possible to expand all quantities in a suitable basis and solve numerically the resulting matrix-eigenvalue equation. As an alternative, we can perform a Laurent expansion of the response function around the excitation energy... 

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