Laurent series

Let the function/be analytic throughout an annular domain z - ZqI Ri, and let C denote any positive oriented simple closed contour around Zq and lying in that domain. Then at each point z in the domain, /(z) has the series representation 

The complex number that is the coefQcient of l/(z - Zq) (i.e., b, of the Laurent series) is known as the residue of/at the isolated singular point Zq- The principal part of/is the portion of the series involving negative exponents. If b = 0 for all m, then /is said to have a pole of order m. If m = 1, it is called a simple pole. If m = o°,/is said to have an essential singularity. If m = 0,/is said to have a removable singularity. 

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For a range of potential in which the interfacial charge is relatively small, the reciprocal of the interfacial electric capacity, C, of metal electrodes has conventionally been represented by a Laurent series with respect to the Debye length L-o of aqueous solution as shown in Eqn. 5-25 [Schmickler, 1993] ... 

Tavlor/Laurent Series. The Taylor (Laurent) series capability is very impressive. Below we ask for the first 15 terms of the series of (Dl) about the point X 0. Notice that KACSYKA computes this expression in less than 1/2 CPU second. 

The program can also compute Taylor (Laurent) series in several variables ... 

Apart from the truncation of the Laurent series, two further approximations are necessary ... 

Definition A.13 (Residue) If zq is an isolated singular point of f(z), then there exists a Laurent series... 

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 the generic case / / 0 it is reasonable to introduce a generating function in the form of the Laurent series of an auxiliary variable z... 

In the same way, in the complex plane, an operator f(z) can be associated with T(z). This operator can be expanded in the form of Laurent series... 

The variational estimate is consistent with our previous development because from the two first terms of the Laurent series (3.32) we obtain... 

The energy dependence of the logarithmic derivative (2.3) may be established by numerical integration of the Schrodinger equation (1.17) and evaluation of (2.3) over a range of energies. An example of this energy dependence is shown schematically in Fig.2.6, and below we shall show how the v th period of D (E) may be approximated by a Laurent series. 

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