Closed-form Integral Representation Research Articles

Electromagnetic extension of the Dory–Guest–Harris instability as a benchmark for Vlasov–Maxwell continuum kinetic simulations of magnetized plasmas

A closed-form integral representation of the electromagnetic dispersion relation for plasma waves propagating perpendicular to a magnetic field is derived. Growth rates and oscillation frequencies are calculated for several cases of the Dory–Guest–Harris instability and compared with those calculated from the usual electrostatic version of the dispersion relation. The comparisons show that the electromagnetic treatment more accurately identifies unstable configurations in plasmas with high beta, where the electrostatic dispersion relation predicts stability. Continuum kinetic simulations using the Washington approximate Riemann plasma framework confirm the theoretical calculations. The electromagnetic extension of the Dory–Guest–Harris instability provides a new benchmark problem for testing continuum kinetic simulations using the Vlasov–Maxwell plasma model, including for other numerical treatments such as particle-in-cell methods.

Landau quantized dynamics and spectrum of the diced lattice

In this work the role of magnetic Landau quantization in the dynamics and spectrum of diced lattice charge carriers is studied in terms of the associated pseudospin 1 Green’s function. The equations of motion for the 9 matrix elements of this Green’s function are formulated in position/frequency representation and are solved explicitly in terms of a closed form integral representation involving only elementary functions. The latter is subsequently expanded in a Laguerre eigenfunction series whose frequency poles identify the discretized energy spectrum for the Landau-quantized diced lattice as ( is the characteristic speed for the diced lattice) which differs significantly from the nonrelativistic linear dependence of ϵ n on B, and is similar to the corresponding dependence of other Dirac materials (graphene, group VI dichalcogenides).

Generalized Volterra functions, its integral representations and applications to the Mathieu-type series

In this paper we introduce the new class of generalized Volterra functions. We prove some integral representations for them via Fox–Wright H-functions and Meijer G-functions. From positivity conditions on the weight in these representations, we found sufficient conditions on parameters of the generalized Volterra function to prove its complete monotonicity. As applications, a Turán type inequality for generalized Volterra functions is derived, infinite integral of some special functions are expressed in terms of the generalized Volterra functions and closed-form integral representations for a family of convergent Mathieu-type series defined in terms of generalized Volterra functions are established.

Integral Representation for Bessel’s Functions of the First Kind and Neumann Series

A Fourier-type integral representation for Bessel’s functions of the first kind and complex order is obtained by using the Gegenbauer extension of Poisson’s integral representation for the Bessel function along with a suitable trigonometric integral representation of Gegenbauer’s polynomials. By using this representation, expansions in series of Bessel’s functions of various functions which are related to the incomplete gamma function can be obtained in a unified way. Neumann series are then considered and a new closed-form integral representation for this class of series is given. The density function of this representation is the generating function of the sequence of coefficients of the Neumann series on the unit circle. Examples of new closed-form integral representations of special functions are thus presented.

Landau quantized dynamics and spectra for group-VI dichalcogenides, including a model quantum wire

This work is concerned with the derivation of the Green’s function for Landau-quantized carriers in the Group-VI dichalcogenides. In the spatially homogeneous case, the Green’s function is separated into a Peierls phase factor and a translationally invariant part which is determined in a closed form integral representation involving only elementary functions. The latter is expanded in an eigenfunction series of Laguerre polynomials. These results for the retarded Green’s function are presented in both position and momentum representations, and yet another closed form representation is derived in circular coordinates in terms of the Bessel wave function of the second kind (not to be confused with the Bessel function). The case of a quantum wire is also addressed, representing the quantum wire in terms of a model one-dimensional δ(x)-potential profile. This retarded Green’s function for propagation directly along the wire is determined exactly in terms of the corresponding Green’s function for the system without the δ(x)-potential, and the Landau quantized eigenenergy dispersion relation is examined. The thermodynamic Green’s function for the dichalcogenide carriers in a normal magnetic field is formulated here in terms of its spectral weight, and its solution is presented in a momentum/integral representation involving only elementary functions, which is subsequently expanded in Laguerre eigenfunctions and presented in both momentum and position representations.

Liquid toroidal drop in compressional flow with arbitrary drop-to-ambient fluid viscosity ratio

Existing experiments show that a sufficiently fat toroidal drop freely suspended in another liquid shrinks towards its centre to form a spherical drop. However, recent simulations reveal that if a liquid torus with circular cross section is embedded in a compressional same-viscosity flow that acts to expand the torus, then depending on the torus radius R and a capillary number Ca characterizing the balance between the viscous forces and the interfacial tension, the torus may either coalesce, expand indefinitely or attain a stationary shape. For each Ca less than 0.2, there is a single value of R , called the critical radius, for which the torus attains the stationary shape. Here, the drop-to-ambient fluid viscosity ratio, λ, is assumed to be arbitrary. The corresponding two-phase Stokes flow problem is solved for a liquid toroidal drop with circular cross section in terms of stream functions in the toroidal coordinates. When λ=1, the stream functions admit a closed-form integral representation for a drop of arbitrary axisymmetric shape. ‘Stationary’ circular tori minimize a certain measure of the normal velocity over the interface, and as in the case of λ=1, their radii are expected to predict the critical ones for arbitrary λ and Ca in a certain range (e.g. for Ca<0.2 when λ=1). Streamlines about ‘stationary’ circular tori are analysed for various Ca and λ.

On an integral representation of the Neumann—Tricomi problem for the Lavrent’ev–Bitsadze equation

We use the spectral method to prove the existence and uniqueness of a regular solution of the Neumann—Tricomi problem for the Lavrent’ev—Bitsadze equation in a half-strip. The Frankl condition is posed on the type change line of the equation. We obtain a closed-form integral representation of the solution of this problem.

Gellerstedt problem for a differential-difference equation of mixed type with advanced-retarded multiple deviations of the argument

We consider the Gellerstedt problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator in the leading part and with advanced-retarded multiple deviations of the argument in the derivatives and the function. We prove the uniqueness theorem for the problem without restrictions on the deviation value. The problem is uniquely solvable. We derive closed-form integral representations of the solutions.

Analytical solution for a phreatic groundwater fall: the Riesenkampf and Numerov solutions revisited

Steady, two-dimensional Darcian flow in a homogeneous isotropic unconfined aquifer, bounded from below by a rectangular wedge representing bedrock, is studied by the theory of holomorphic functions. A triangle of the complex potential domain is mapped onto a circular triangle in the hodograph plane with the help of an auxiliary variable. A full potential theory results in closed-form integral representations for the complex potential and complex velocity, from which the flow rate and free surface are calculated using computer algebra built-in functions. This solution, uniformly valid in the whole flow domain, is compared with simpler approximate ones, retrieved from an analytical archive. Two flow zones are distinguished: a tranquil subdomain where the Dupuit-Forchheimer approximation is suitable and a nappe (a subdomain with a rapidly changing Darcian velocity and steep slope of the phreatic surface) where the Numerov or Polubarinova-Kochina solutions, in terms of the full potential model, are available. Approximations in the two zones are conjugated by matching the positions of the water table and the flow rates, which eventually agree well with the obtained comprehensive solution.

Rational approximation formula for Chandrasekhar’s H-function for isotropic scattering

We first establish a simple procedure to obtain with 11-figure accuracy the values of Chandrasekhar's H-function for isotropic scattering using a closed-form integral representation and the Gauss-Legendre quadrature. Based on the numerical values of the function produced by this method for various values of the single scattering albedo and the cosine of the azimuth angle of the direction of radiation emergent from or incident upon a semi-infinite scattering-absorbing medium, we propose a rational approximation formula, which allows us to reproduce the correct values of the H-function within a relative error of 2.1/100000 without recourse to any iterative procedure or root-finding process.

A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model

We consider the problem of pricing European exotic path-dependent derivatives on an underlying described by the Heston stochastic volatility model. Lipton has found a closed form integral representation of the joint transition probability density function of underlying price and variance in the Heston model. We give a convenient numerical approximation of this formula and we use the obtained approximated transition probability density function to price discrete path-dependent options as discounted expectations. The expected value of the payoff is calculated evaluating an integral with the Monte Carlo method using a variance reduction technique based on a suitable approximation of the transition probability density function of the Heston model. As a test case, we evaluate the price of a discrete arithmetic average Asian option, when the average over n = 12 prices is considered, that is when the integral to evaluate is a 2 n = 24 dimensional integral. We show that the method proposed is computationally efficient and gives accurate results.

Spectral efficiency of optical direct detection

The spectral efficiency (channel capacity) of the optical direct-detection channel is studied. The modeling of the optical direct-detection channel as a discrete-time Poisson channel is reviewed. Closed-form integral representations for the entropy of random variables with Poisson and negative binomial distributions are derived. The spectral efficiency achievable with an arbitrary input gamma density is expressed in closed integral form. Simple, nonasymptotic upper and lower bounds to the channel capacity are computed. Numerical results are presented and compared with previous bounds and approximations.

Closed form representation for a projection onto infinitely-dimensional subspace spanned by Coulomb bound states

The closed form integral representation for the projection onto the subspace spanned by bound states of the two-body Coulomb Hamiltonian is obtained. The projection operator onto the n2-dimensional subspace corresponding to the nth eigenvalue in the Coulomb discrete spectrum is also represented as the combination of Laguerre polynomials of nth and (n − 1)th order. The latter allows us to derive an analogue of the Christoffel–Darboux summation formula for the Laguerre polynomials. The representations obtained are believed to be helpful in solving the breakup problem in a system of three charged particles where the correct treatment of infinitely many bound states in two-body subsystems is one of the most difficult technical problems.

Analysis of eddy-current losses over conductive substrates with applications to monolithic inductors and transformers

In this paper, a closed-form integral representation for the eddy-current losses over a conductive substrate is presented. The results are applicable to monolithic inductors and transformers, especially when such structures are realized over an epitaxial CMOS substrate. The technique is verified against measured results from 100 MHz to 14 GHz for spiral inductors.

Transient dynamic response of piezoelectric bodies subjected to internal electric impulses

The response of piezoelectric bodies disturbed by internal electric sources is investigated using integral representations of Greens functions and their derivatives. The emphasis of the article is placed on transient dynamic phenomena promoted by high intensity electric discharges applied over very short intervals of time. To that end closed form integral representations of the electro-elastic variables are derived and applied to investigate the behavior of transversely isotropic piezoelectric ceramics. In particular, the characteristics of the induced stress and electric field components are investigated numerically for the case of charge impulses of triangular shape.

Thermodynamic Green's-function theory of quantum magnetic field effects in the dynamic, nonlocal longitudinal dielectric-response properties of a bounded solid-state plasma.

The analysis of dynamic nonlocal longitudinal dielectric response properties of a slab of quantum plasma in a magnetic field H (perpendicular to the plane slab faces) is carried out here with use of a thermodynamic Green's-function formulation of the random-phase approximation (RPA). The magnetic field Green's function for the slab incorporates magnetic field effects in terms of a closed-form integral representation, and the boundary condition of specular reflection is imposed in two alternative ways, in terms of (1) a partial eigenfunction expansion, and (2) an image series of infinite-space Green's functions. The RPA integral equation is solved for the direct slab dielectric function subject to Landau quantization, with results expressed in terms of the density perturbation response function scrR=\ensuremath{\delta}\ensuremath{\rho}/\ensuremath{\delta}V which depends on both z and z' because of the lack of translational invariance perpendicular to the slab faces (parallel to H=Hz^). Correspondingly, scrR depends on two conjugate wave-vector transform variables ${q}_{z}$ and ${q}_{z}^{\mathcal{'}}$ which are interpreted as indices for rows and columns of scrR regarded as a matrix. The magnetic field dependencies and nonlocality of both the diagonal and nondiagonal elements of scrR are thoroughly examined here. Applications of this work to the Landau quantized nonlocal slab surface-plasmon dispersion relation are discussed.