Dirac Delta Functions in the Laplace‐Type Expansion of rnYlm(θ, φ)

1969 ◽  
Vol 51 (6) ◽  
pp. 2359-2362 ◽  
Author(s):  
Kenneth G. Kay ◽  
H. David Todd ◽  
Harris J. Silverstone
Keyword(s):  
Dirac Delta ◽  
Delta Functions ◽  
Laplace Type

scholarly journals Electrostatic and magnetostatic fields of point dipoles revisited

2019 ◽  
Vol 65 (1) ◽  
pp. 71 ◽  
Author(s):  
Y. Muniz ◽  
Anderson José Fonseca ◽  
C. Farina
Keyword(s):  
Magnetic Dipole ◽  
Electric Dipole ◽  
Alternative Procedure ◽  
Dirac Delta ◽  
Delta Functions ◽  
Point Electric Dipole

After reviewing how the Dirac delta contributions to the electrostatic and magnetostatic fields of a point electric dipole and a point magnetic dipole are usually introduced, we present an alternative procedure for obtaining these terms based on a regularization prescription similar to that used in the computation of the transverse and longitudinal delta functions. We think this method may be useful for the students in other analogous calculations.

Review of basic quantum physics

Quantum 20/20 ◽  
2019 ◽  
pp. 1-20
Author(s):  
Ian R. Kenyon
Keyword(s):  
Quantum Physics ◽  
Particle Density ◽  
Electromagnetic Coupling ◽  
Black Body ◽  
Black Body Radiation ◽  
Gauge Transformations ◽  
Dirac Delta ◽  
Wave Particle Duality ◽  
Delta Functions ◽  
Lorentz Invariants

Basic experimental evidence is sketched: the black body radiation spectrum, the photoeffect, Compton scattering and electron diffraction; the Bohr model of the atom. Quantum mechanics is reviewed using the Copenhagen interpretation: eigenstates, observables, hermitian operators and expectation values are explained. Wave-particle duality, Schrödinger’s equation, and expressions for particle density and current are described. The uncertainty principle, the collapse of the wavefunction, Schrödinger’s cat and the no-cloning theorem are discussed. Dirac delta functions and the usage of wavepackets are explained. An introduction to state vectors in Hilbert space and the bra-ket notation is given. Abstracts of special relativity and Lorentz invariants follow. Minimal electromagnetic coupling and the gauge transformations are explained.

Variational Modal Identification of Conservative Nongyroscopic Systems

1989 ◽  
Vol 111 (2) ◽  
pp. 160-171 ◽  
Author(s):  
L. Silverberg ◽  
S. Kang
Keyword(s):  
Longitudinal Vibration ◽  
Modal Identification ◽  
System Response ◽  
Bending Vibration ◽  
Free System ◽  
Dirac Delta ◽  
Vibration Problem ◽  
Identification Method ◽  
Delta Functions ◽  
Admissible Functions

A new modal identification method for Conservative Nongyroscopic Systems is proposed. The modal identification method is formulated as a variational problem in which stationary values of a functional quotient are sought. The computation of the functional quotient is carried out using a set of admissible functions defined over the spatial domain of the system. Measurements of the free system response at discrete points are carried out using any combination of displacements, velocities, and/or accelerations. Three types of admissible functions have been considered—global functions, spatial Dirac-delta functions, and finite element interpolation functions. The variational modal identification method is applied to a pure bending vibration problem, to a pure longitudinal vibration problem, and to a combined bending and longitudinal vibration problem. The effectiveness of the variational modal identification method using different sets of admissible functions is examined.

Dirac delta functions

2015 ◽  
pp. 489-492
Author(s):  
J. M. Carpenter ◽  
C.-K. Loong ◽  
Marie-Louise Saboungi
Keyword(s):  
Dirac Delta ◽  
Delta Functions

Tutorial/Artical didactique: A high-accuracy mathematical and numerical method for Fourier transform, integral, derivatives, and polynomial splines of any order

1998 ◽  
Vol 76 (9) ◽  
pp. 659-677 ◽  
Author(s):  
N Beaudoin
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Computation Time ◽  
Polynomial Splines ◽  
Dirac Delta ◽  
Optimum Parameters ◽  
Complex Functions ◽  
Discrete Counterpart ◽  
Delta Functions ◽  
Fifth Order

From few simple and relatively well-known mathematical tools, an easily understandable, though powerful, method has been devised that gives many useful results about numerical functions. With mere Taylor expansions, Dirac delta functions and Fourier transform with its discrete counterpart, the DFT, we can obtain, from a digitized function, its integral between any limits, its Fourier transform without band limitations and its derivatives of any order. The same method intrinsically produces polynomial splines of any order and automatically generates the best possible end conditions. For a given digitized function, procedures to determine the optimum parameters of the method are presented. The way the method is structured makes it easy to estimate fairly accurately the error for any result obtained. Tests conducted on nontrivial numerical functions show that relative as well as absolute errors can be much smaller than 10-100, and there is no indication that even better results could not be obtained. The method works with real or complex functions as well; hence, it can be used for inverse Fourier transforms too. Implementing the method is an easy task, particularly if one uses symbolic mathematical software to establish the formulas. Once formulas are worked out, they can be efficiently implemented in a fast compiled program. The method is relatively fast; comparisons between computation time for fast Fourier transform and Fourier transform computed at different orders are presented. Accuracy increases exponentially while computation time increases quadratically with the order. So, as long as one can afford it, the trade-off is beneficial. As an example, for the fifth order, computation time is only ten times greater than that of the FFT while accuracy is 108 times better. Comparisons with other methods are presented.PACS Nos.: 02.00 and 02.60

scholarly journals Integral Equations of Non-Integer Orders and Discrete Maps with Memory

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1177
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Exact Solutions ◽  
Integral Equations ◽  
Fractional Integral ◽  
Periodic Sequence ◽  
Fractional Integrals ◽  
Dirac Delta ◽  
Discrete Maps ◽  
Delta Functions ◽  
First Time ◽  
With Memory

In this paper, we use integral equations of non-integer orders to derive discrete maps with memory. Note that discrete maps with memory were not previously derived from fractional integral equations of non-integer orders. Such a derivation of discrete maps with memory is proposed for the first time in this work. In this paper, we derived discrete maps with nonlocality in time and memory from exact solutions of fractional integral equations with the Riemann–Liouville and Hadamard type fractional integrals of non-integer orders and periodic sequence of kicks that are described by Dirac delta-functions. The suggested discrete maps with nonlocality in time are derived from these fractional integral equations without any approximation and can be considered as exact discrete analogs of these equations. The discrete maps with memory, which are derived from integral equations with the Hadamard type fractional integrals, do not depend on the period of kicks.

scholarly journals Ordinary differential equations with delta function terms

2012 ◽  
Vol 91 (105) ◽  
pp. 125-135 ◽  
Author(s):  
Marko Nedeljkov ◽  
Michael Oberguggenberger
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Delta Function ◽  
Limiting Distribution ◽  
Nonlinear Ordinary Differential Equations ◽  
Smooth Solutions ◽  
Dirac Delta ◽  
Delta Functions ◽  
Derivatives Of

This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family of smooth solutions. Conditions on the nonlinear terms, relating to the order of the derivatives of the delta function part, are established so that the regularized solutions converge to a limiting distribution.

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