Poles simple

When Eq. (230) is replaced in Eq. (228), the density correlation function is found to have two simple poles,... 

In the expansion in powers of u again all higher order poles have to cancel so that there survive only those simple poles multiplied by u. As a result the power series for 1 fv(u) is independent of e. 

Here we have taken the two wires to have the same electron density and strength of the interactions. For not very strong interactions, a 1, the last term in Eq. (6) can be disregarded, away from the regime of the zero-bias anomaly. [7] The integrand in Eq. (6) then has two simple poles yielding... 

From Eq. (C.2) we conclude that the square integrable solution / contains the independent solution spectral function p(E) to be used in the completeness relation and the eigenfunction expansion. The former gives... 

When x < t, analogous reasoning shows that the contour of integration must be closed to the left, and the simple pole at s = 0 and branch points at s = — 1 and at s = — deolf foY will provide a nonzero contribution to V. Thus some disturbance always propagates with the velocity... 

The inverse Laplace transform of the above expression can be found by noting that there are simple poles at... 

Gp p, p r) at p = 0 have only simple poles and cuts as function of r- In perturbation theory this property holds true for the zero-order tcum, a simple pole being found for r = 0, The first order correction, however, at T = 0 has a pole of second order, due to the two factors of Gqj (p,t). This signals that the interaction wants to shift the pole according to the mechanism... 

The appearance of ir/ in the denominators here defines the analytical properties of this function The fact that / (cy) is analytic on the upper half of the complex m plane and has simple poles (associated with the spectrum of Hq") on the lower half is equivalent to the casual nature of its Fourier transform—the fact that it vanishes for Z < 0. An interesting mathematical property follows. For any function / (cy) that is (1) analytic in the half plane Recy > 0 and (2) vanishes fast enough for cy - oo we can write (see Section 1.1.6)... 

Evidently, in Eq. (3) the term associated with the nuclear Coulomb repulsion, A/R, is added by Engelke. Then /3 = AlRe. This term has a simple pole at R = 0. If, moreover, we add the centrifugal potential of the type B/R2 and write the modified Coulomb-centrifugal-subtracted D-expansion as... 

There are two simple poles, ib and -ib, on the imaginary axis. An appropriate contour for this integral is shown in Figure 2.3. 

On the other hand, S is independent of a and b. The problem is that, in general, it does not have a simple pole structure. It is always possible to write S as a. sum of poles ... 

One proceeds similarly for the other two divergent functions. In the case of the massless photons the propagator must have a simple pole at = 0,... 

This can be proven, e.g., by reducing to n = 2 and in this case showing that in some neighbourhood JJ of H, there is a meromorphic 2-form u on U, with simple poles on U7 f) U, such that... 

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