Residue theorem

Let/(z) be analytic inside and on a simple closed contour C, except for a finite number of singular 

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The main goal of the pn ocedure adopted to solve the propagator equation is to find the pole(s) of interested and the residues associated to it. Since G (E) has single isolated poles then the residue theorem can be applied to yield... 

We have closed the contour C by means of a semicircle of infinite radius in the lower half plane. The integrand is zero on this half circle because r is positive. In the closed contour thus obtained, we have applied the residue theorem. The only contributions to be retained come from the poles at z 0 indeed, the poles of the operators Yg/ -n) are situated in the lower half plane and give terms affected by an exponential which vanishes for r —> oo. 

Let us now substitute this expression in Eq. (93) where we pass to the limit t- - oo. Since t > 0, let us close the contour C by means of a large semicircle in the lower half plane and apply the residue theorem. The result is... 

We can at present close the contour C and apply the residue theorem, which gives (see Eq. 83) ... 

The explicite expression for (49) can be obtained by using the Cauchy residue theorem. For this aim, the strength function is recasted as a sum of V residues for the poles z = Since the sum of all the residues (covering all the poles) is zero, the residues with z = Eojp (whose calculation is time consuming) can be replaced by the sum of residies with z = u) E f(A/2) and z = Esph whose calculation is much less expensive (see details of the derivation in [8]). 

When we inspect this equation we realize that there are two functional problems. First, there is one equation per element with two unknowns. Second, we are using a linear approximation for the temperature however, we have second spatial derivatives of temperature. The first problem is solved by using Garlekin s method of weighted residuals (Theorem (9.1.1)). 

The residues theorem allows treating the resolvent as a formal solution of the eigenvector/eigenvalue problem. Indeed, taking a contour integral over any path Ct enclosing each of the poles one gets ... 

According to the residue theorem applied to the k" integral the scattering is determined by the poles of the partial T-matrix element in the complex k" plane. The existence and positions of the poles are of course determined by the details of the potential V, but we will assume that there is a pole corresponding to complex energy Cr — iTr. The magnitude of the partial T-matrix element varies rapidly with values of E near the pole and we can consider er as the resonance energy. For the cross section we need only consider the on-shell partial T-matrix element... 

Theorem A.4 (Cauchy s Residue Theorem) If f z) is analytic in a simply connected domain D, except at a finite number of singular points zi,...,Zk and if C is a simple positively oriented (counterclockwise) closed contour that lies in D, then... 

The denominator of the three-vertex term, 1 — Ay — Aik — Ajk — 2Ayk = 0, reduces to a cubic polynomial in z, which can be solved to yield three roots, which are in turn used to evaluated the contour integral using the residue theorem. 

If there are multiple poles, then the appropriate version of the residue theorem needs to be applied, viz ... 

A time periodic solution of the set of equations 15.17a to 15.17d with a period 2nr is easily derived by applying the residue theorem to the general inversion integral of the Laplace transform solution of these equations [15]. It can be written as ... 

For a long time it has been the only way, by the use of the Laplace method. Such a method is based on the representation in the complex plane of the confluent hypergeometric functions and the use the residues theorem. But it needs the approximation Z2a2 C ft2, which allows one to replace in (12.16)... 

One of the many applications of the theory of complex variables is the application of the residue theorem to evaluate definite real integrals. Another is to use conformal mapping to solve boundary-value problems involving harmonic functions. The residue theorem is also very useful in evaluating integrals resulting from solutions of differential equations by the method of integral transforms. 

Applying the residue theorem to the contour integral gives... 

G e ) is per dehnition zero. In the case of nondegenerate roots, according to the residue theorem only the Hrst term in the Taylor expansion of the denominator contributes. Applying the dennition of the 6 function Eq.(2.152) results in Eq.(2.197). 

This is called the residue of /(z) and plays a very significant role in complex analysis. If a function contains several singular points within the contour C, the contour can be shrunken to a series of small circles around the singularities Zn, as shown in Fig. 13.6. The residue theorem states that the value of the contour integral is given by... 

Now we see the Cauchy Residue theorem gives the compact result... 

Actually, for the concerned function two poles are identified in the complex plane of Figure 3.13, producing the associated integrals to be solved according with the Cauchy residues theorem with application to the single or to multiple poles as well ... 

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