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Method of Finite Differences and Self-Energy of the Leads
Published in Vinod Kumar Khanna, Introductory Nanoelectronics, 2020
Cauchy’s residue theorem states that: For a function f(z) which is analytic on a simple positively oriented closed contour γ and everywhere inside this contour, with exception of the finite number of points z1, z2, z3, …zn inside the above contour, the line integral of the function f(z) around the closed contour γ is 2πi times the sum of residues of f(z) at the points: |∮f(z)dz|γ=2πiΣk=1nRes(f,zk)
Complex Analysis
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
Remark 71. The evaluation of integrals using the residue theorem depends on the determination of residues at singular points. The residue theorem should be used to evaluate integrals of those functions and finding those residues at singular points is not tedious.
Remarks on the inverse problem for an energy-dependent hamiltonian
Published in Applicable Analysis, 2023
Pierre Chau Huu-Tai, Bernard Ducomet
Let us consider the family of small circles clockwise surrounding the spectral singularities for : with small enough and define the function Using residue theorem we see that A straightforward computation gives now where the are polynomials of degree , and because we get the estimate for positive constants .
Inverse scattering problem for detecting a defect in a magnetoelastic layer
Published in Inverse Problems in Science and Engineering, 2021
Khaled M. Elmorabie, Rania Yahya
For constructing the particular solutions of the differential equations system (26), the Fourier transform is applied two times over the variables , with the parameters s, and β respectively leads to linear algebraic system as follows: By using Cramer's rule, the solution to this algebraic system may be found. Therefore the particular solutions after applying the inverse Fourier for β, and after some routines calculations, take the form According to the complex residue theorem and the following identity, one can obtain the final solutions of the integral expressions in Eqaution (31) as follows: where is a Hankel function of the first kind and zero order, and
A central limit theorem for periodic orbits of hyperbolic flows
Published in Dynamical Systems, 2021
Stephen Cantrell, Richard Sharp
We first note that for , with , we have the trivial estimate We will prove the result using contour integration. Choose numbers such that, for , lies in the interior of the rectangle with vertices at , , and . where . Furthermore, decreasing δ if necessary, we may assume that this rectangle lies within the region of analyticity described in Proposition 3.1. Set , and where . Take , where β is the same as in Proposition 3.1. By the Residue Theorem we may write where Γ is the contour consisting of the straight lines connecting the points , , , , , , , and .