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Existence Theorems and Special Functions
Published in L.M.B.C. Campos, Singular Differential Equations and Special Functions, 2019
as the irregular integral of the second kind with a combined essential singularity and power-type branch-point. It has been shown that if the linear second-order differential equation (9.641c) with coefficients having an isolated singularity (9.641a, b) has (standard CCCLI) equal eigenvalues (9.663a), or equivalently (9.649a) indices (9.663b) differing by an integer (9.652d, e)≡(9.662a), then the general integral (9.663b):λ+=λ−⇔a+−a−∈|N:yx=C+y+x+C−y¯−x,
Modal Properties of Layered Metamaterials
Published in Filippo Capolino, Theory and Phenomena of Metamaterials, 2017
Paolo Baccarelli, Paolo Burghignoli, Alessandro Galli, Paolo Lampariello, Giampiero Lovat, Simone Paulotto, Guido Valerio
In this section properties of leaky modes in grounded metamaterial slabs are discussed. As is well known, leaky modes are complex modes, that is, their propagation wave number kz = β − jα is complex. A nonzero attenuation constant α is present, also in lossless structures, due to radiation losses associated with the propagation of the mode in a transversely open environment. An important issue in connection with leaky modes is their spectral character, which is established by their behavior at infinity in the transverse plane: if a leaky mode satisfies the radiation condition at infinity, it is termed proper, otherwise improper. This is in turn related to the nature of leaky-mode radiation, which is backward for proper modes and forward for improper modes. We recall here that modal solutions in waveguiding structures correspond to pole singularities of the waveguide Green’s function in the complex plane of the relevant spectral variable (for grounded slabs, such a variable can be kz for 1D modal propagation along the z axis or kρ for 2D propagation along the radial axis ρ); the spectral Green’s function has also square-root branch points at the wavenumbers ±k0 of the air medium, and the proper or improper character of a mode is related to the location of the corresponding pole in different Riemann sheets with respect to that branch point [28].
Mathematical Preliminaries
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
Each branch is single valued, but to keep the function single valued, we must set up an artificial barrier that joins the origin and infinity. This artificial barrier is called a branch line or a branch cut, and this barrier should not be crossed for each branch of solutions. For this case, the origin, for which a multivalued function appears when we go around this point a complete circuit, is called a branch point. As illustrated in Figure 1.14, we have chosen the branch cut along the positive xaxis, but actually any other branch cut connecting the origin and infinity along any orientation can be selected.
Constituents of electromagnetic 2-D layered media Green's functions for all material types and radiation conditions
Published in Waves in Random and Complex Media, 2023
Even though the closed-form representations in (2) are used for the direct fields in the source layer-i (by setting since ) without taking any integrations, the SIP in Figure 2 would have been used to obtain the same results by the well-known formula in the most general form: When performing the spectral to spatial domain transform by (17), the branch point and associated branch cut of the source layer are only observed on the complex -plane when the observation point is in the source layer-i. Because the direct terms generated by the source are not included in the integrands defined in (11). When the observation point is on a different layer than layer-i or the direct term is included in the integrands, the branch point and the associated branch cut of the source layer would disappear. Even in such cases, the SIP should be defined by assuming that there is a branch point and associated cut at the wavenumber of the source layer-i to include the direct terms in the integrations that are consistent with the boundary condition.
Resummation of the Rayleigh-Schrödinger perturbation series. Vibrational energy levels of the H2S molecule.
Published in Molecular Physics, 2021
As mentioned above, the matrix eigenvalues can be considered as different branches of a single function of the perturbation parameter λ defined on a multi-sheet Riemann surface [19]. Since the function is multi-valued, there exist quadratic branch points that connect two different branches, the so-called Katz branch points [8,20]. Since for real λ the eigenvalues of a Hermitian matrix are real, branch points can come only in conjugate pairs not lying on the real axis. Two branches at the Katz branch points coincide and are continuous, but their derivatives with respect to λ are undefined. As a result, at Katz branch point these two branches are not analytic. Going around this point, one passes from one sheet of the Riemann surface to another. In other words, an analytic continuation along a closed path that completely encircles a Katz branch point, allows the eigenvalue on one sheet to be found from the eigenvalue given on the other. Among possible approximate methods of analytic continuation, multivalued Hermite -Padé algebraic approximants, which reflect the suspected singularity structure, were found to be effective [8,9,14].
A highly accurate analytical solution for the surface fields of a short vertical wire antenna lying on a multilayer ground
Published in Waves in Random and Complex Media, 2018
The consequence of this choice is that the branch point behavior in the upper half of the complex plane is to be attributed to the square roots and . Accordingly, we let