Singularities branch points

Now consider the ground state function E (z). Starting at the origin, trace a path in the z-plane that circles about the point zoi This point is a branch point singularity, which means that a 360° circuit will lead to a new Riemann sheet corresponding to the fimction E > (z). Similarly, following E ) (z) on a path that circles zq2 leads to E (z), and so on. 

We have been able to fit the parameter 6 even more precisely using a modified version of the Fade singularity analysis [10]. Fade, in his original treatise [35], suggested a generalization of his approximants that can explicitly model branch-point singularities. Let P S) and Q(S) be the numerator and denominator polynomials, respectively. Then Eq. (15) can be written in the form of a linear equation in E,... 

Thus, our solution at = 0 corresponds to the solution of the unsealed Schrodinger equation in the limit R — oo. The energy as a function of 1/R is known to have a complicated branch point singularity in this limit [39], which has been attributed [40] to the fact that at infinite R the electron is localized at only one of the protons, while at... 

The asymptotic behavior of the coefficients / and mn is determined by the singularities nearest to the real axis in the complex 0 plane. These singularities are square root and inverse square root branch points at i0 = In A. From this it follows that... 

The theoretical analysis here in the present section clearly indicates that the localized delta function excitation in the physical space is supported by the essential singularity (a —> oo) in the image plane. This is made possible because 4> y, a) does not satisfy the condition required for the satisfaction of Jordan s lemma. As any arbitrary function can be shown as a convolution of delta functions with the function depicting the input to the dynamical system. The present analysis indicates that any arbitrary disturbances can be expressed in terms of a few discrete eigenvalues and the essential singularity. In any flow, in addition to these singularities there can be contributions from continuous spectra and branch points - if these are present. 

The integrand of eq. 4.140 in general use has two types of singularities namely branch points and poles. First we will consider a two-layered medium when the invasion zone is absent. Analysis of zeroes of a determinant of function Ci, as well as calculations shows... 

Singularities of the integrand of function C are in general poles and branch points. Numerical analysis shows that for relatively large values of wavelength A = 2-Kh poles are absent in the upper half-plane where only three branch points are located, namely iki, ik2, and ifcs. 

We will assume that in the upper half-plane of complex variable m there are no singularities except the branch points m = k a and m = k2a. 

Other kinds of approximants can also be used. For example, Olsen et al. [20] analyzed MP series convergence using a 2 X 2 matrix eigenvalue equation [38,39], which implicitly incorporates a square-root branch point. It is of course possible simply to explicitly construct an approximant as an arbitrary function with the singularity structure that E z) is expected to have. We suggest, for example, approximants of the form ... 

This argument identifies the location of a singularity, but it does not elucidate its type, for example, whether it is a pole or a branch point. Since the divergence occurs at the limit r 0, the singularities of E can be characterized by determining the behavior of at small r. In the neighborhood of the origin, (r) formally has the expansion... 

We can rule out a simple square-root branch point for the singularity at the origin since that would lead to half-integer powers of 8 in the asymptotic expansion, Eq. (9) the 6 expansions of the wave-function and of the Hamiltonian do have half-integer powers, but they cancel exactly leaving Eq. (9) as the correct form for the energy ex-... 

Thus, if any contour is drawn so that the branch point is encircled, then multivalued behavior arises. The principle to ensure analyticity is simple branch points cannot be encircled. There is considerable range and scope for choosing branch cuts to ensure analytic behavior. Now, if no other singularities exist in the contour selected, then Cauchy s First Integral theorem is valid, and we denote the new contour C2 as the second Bromwich path, Br2 hence. 

We shall use the second Bromwich path drawn in Fig. 9.13, where the branch point is located at 5 = 0 and the branch cut extends to infinity. We write first that, since no other singularities exist... 

When pole singularities exist, along with branch points, the Cauchy theorem is modified to account for the (possibly infinite) finite poles that exist within the Bromwich Contour hence, E res (F(j)e ) must be added to the line integrals... 

In addition, such a power series expansion is, however, only permitted for analytic, i.e., holomorphic functions and must never be extended beyond a singular point. Since the square root occurring in the relativistic energy-momentum relation Ep of Eq. (11.11) possesses branching points at X = p/nteC = i, any series expansion of Ep around the static nonrela-tivistic limit T = 0 is only related to the exact expression for Ep for non-ultrarelativistic values of the momentum, i.e., t < 1. This is most easily seen by rewriting Ep as... 

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