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Method of Finite Differences and Self-Energy of the Leads
Published in Vinod Kumar Khanna, Introductory Nanoelectronics, 2020
The equation for the Green’s function can be used only when we know the self-energy of the leads. But to evaluate the self-energy of the leads, it is required to determine the Green’s function for an isolated lead. For this calculation, the Schrodinger equation is solved for a wire terminating on one side x = 0 in an infinite potential with a confining potential U(y) in the Y-direction. The Green’s function is obtained by linearly combining the wave functions obtained by solving the Schrodinger equation. For superposition, the Green’s function is expressed as a summation over the product of coefficients Cα with the wave functions. The coefficients Cα are determined by plugging in the Green’s function in summation form into the equation for the Green’s function of the conductor with the leads. The summation over the constant β is replaced with integration by taking into consideration the density of states in one dimension. Contour integration is performed employing Cauchy’s residue theorem. Finally, transformation from the continuous coordinates to discrete lattice gives the required Green’s function of the wire.
Tools For Analysis
Published in James J Y Hsu, Nanocomputing, 2017
This gives an asymptotic expansion rather useful for large x. To prove S∞=π2 $ S_{\infty } = {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0pt} 2} $ , recall that I=∫-∞∞eitdtt=2πi $ I = \int_{ - \infty }^{\infty } {e^{it} \frac{dt}{t} = 2\pi i} $ . Since there is a pole at the origin on the real axis, the contour integration can be found by taking the residue for encircling the pole, while the integrand vanishes on the upper half of the complex plane.
Further Studies of Electromagnetic Waves in Rectangular Geometries
Published in Guillermo Gonzalez, Advanced Electromagnetic Wave Propagation Methods, 2021
The above integrals were originally evaluated in the λ plane. Consider the integral in (8.5.16). This integral can be evaluated using contour integration by closing the path of integration with a large semicircle in the lower-half plane, accounting for the singularities and using residue theory.
Equivalent dynamic stiffnesses and 3D wave propagations of a transversely isotropic elastic ground in rocking and torsional interactions with a harmonically loaded rigid foundation
Published in Mechanics of Advanced Materials and Structures, 2023
Yazdan Hayati, Abolfazl Eslami, Alireza Rahai
It is worthwhile to mention that for thermoelastic materials, the branch points and the Rayleigh pole are complex-valued numbers due to thermo-mechanical coupling [39–41], while for elastic materials studied in this paper, the branch points and the Rayleigh pole are pure real numbers which are located exactly on the common path of integration which is the real axis in the complex plane. Therefore, for elastic materials considered in this paper, the branch points produce weak singularities on the path of integration which can be resolved by implementing several features and commands of Mathematica software as discussed above. However, the Rayleigh pole is a simple pole and produces a strong singularity on the path of integration of the function which is the kernel function for rocking oscillation of the foundation; therefore it is required to use the contour integration and residue theorem to evaluate the related integral successfully. In this case, as explained by Pak [42] and Rahimian et al. [30] and depicted in the Figure 2, the path of integration must be modified by a small semi-circle above the Rayleigh pole and the semi-infinite integral of the kernel function must be evaluated numerically using the contour integration and the residue theorem.
A unified theory for brittle and ductile shear mode fracture
Published in Philosophical Magazine, 2019
To calculate the stress field (6) we may use the contour integration in the complex plane [19]. The function which generates this stress field is:It is interesting to note that this generator is the analytic continuation of the density (4) in the complex plane. It is convenient to introduce a second function :For a mode II crack, we obtain the following expression for the components of the stress tensor:In these equations, the dependence on the position is omitted for simplicity. For a mode III crack, the non-zero components of the stress tensor are
Inverse problems for the matrix Sturm–Liouville equation with a Bessel-type singularity
Published in Applicable Analysis, 2018
The paper is organized as follows. In Section 2, we set the boundary value problem for Equation (1) and formulate inverse problems. We also provide asymptotic formulas, obtained in paper [24], and other preliminaries. In Section 3, the inverse problem by the Weyl matrix is studied. We derive the main equation of the inverse problem by the contour integration in the complex plane of the spectral parameter. Then, we construct the special Banach space of bounded continuous functions, and establish the unique solvability of the main equation in . We provide a constructive algorithm for the solution of the inverse problem, based on the main equation. In Section 4, the inverse problem by the discrete spectral data, which consists of eigenvalues and weight matrices, is studied. We transform the main equation into the linear equation in the Banach space of infinite matrix sequences with the special norm, and show that the resulting equation is uniquely solvable. We develop an algorithm for solution of the inverse problem. Although the considered inverse problems are equivalent, the algorithm by the spectral data is more convenient for numerical solution.