Complex integration poles

This integral is most easily done by complex integration, where the poles of the integrand are a> = zui with a>i = Jcal — y . It leads to... 

FIGURE 3.13 The complex integration contours resulting in the poles I and II and in the closed integral between and turning points of atomic hydrogenic action in the semiclassical quantification. 

Recognizing in the last integral a complex integral, it can be solved by identification of poles through the complex equation... 

The integral (2) of course diverges on the resonance, but one obtains a finite solution, when the poles of the integrant are evaded by complex integration. Using (1) and (4) it follows that... 

This integral is most easily done by complex integration, where the poles of the... 

If the real part of a complex pole is zero, then p = coj. We have a purely sinusoidal behavior with frequency co. If the pole is zero, it is at the origin and corresponds to the integrator 1/s. In time domain, we d have a constant, or a step function. 

To eliminate offset, we need Section 9.2.3. With an added state due to integration, we have to add one more closed-loop poles. We choose it to be -1, sufficiently faster than the real part of the complex poles. The statements are ... 

In the integration for the current expression of Eq. (89) in the complex lower half plane, only the area around the poles where co ek contributes... 

We estimate the integral in the complex f plane. See Fig. A.l for the path for integration and the poles. Notice that the integrand has poles of second order at f = in n +1). Finally, we obtain the following ... 

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... 

Resonances are not truly bound states, but they are interpreted as metastable states. Because of the boundary conditions of resonances, the problem is not Hermitian even if the Hamiltonian is (that is, for square integrable eigenfunctions). Resonances are characterized by complex eigenvalues (complex poles of the scattering amplitude)... 

In this subsection, it has been assumed that the Hamiltonian H—and hence also the Liouvillian L—does not contain the time t explicitly, and the evolution operators S and are then given by the simple formulas (1.12) and (1.13), respectively. In such a case, it should be remembered that the resolvents R(z) and A(z) are the Fourier transforms of the evolution operators S and S, respectively, provided that one introduces integration contours in the complex plane which enclose all poles of the resolvents in a positive sense. 

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