Contour Integral Representation Research Articles

On the derivative of the Legendre function of the first kind with respect to its degree

We study the derivative of the Legendre function of the first kind, Pν(z), with respect to its degree ν. At first, we provide two contour integral representations for ∂Pν(z)/∂ν. Then, we proceed to investigate the case of [∂Pν(z)/∂ν]ν=n, with n being an integer; this case is met in some physical and engineering problems. Since it holds that , we focus on the sub-case of n being a non-negative integer. We show that where Rn(z) is a polynomial in z of degree n. We present alternative derivations of several known explicit expressions for Rn(z) and also add some new. A generating function for Rn(z) is also constructed. Properties of the polynomials Vn(z) = [Rn(z) + (−1)nRn(−z)]/2 and Wn−1(z) = −[Rn(z) − (−1)nRn(−z)]/2 are also investigated. It is found that Wn−1(z) is the Christoffel polynomial, well known from the theory of the Legendre function of the second kind, Qn(z). As examples of applications of the results obtained, we present non-standard derivations of some representations of Qn(z), sum to closed forms some Legendre series, evaluate some definite integrals involving Legendre polynomials and also derive an explicit representation of the indefinite integral of the Legendre polynomial squared.

Surface Casimir densities on a spherical brane in Rindler-like spacetimes

Abstract The vacuum expectation value of the surface energy–momentum tensor is evaluated for a scalar field obeying Robin boundary condition on a spherical brane in ( D + 1 ) -dimensional spacetime Ri × S D − 1 , where Ri is a two-dimensional Rindler spacetime. The generalized zeta function technique is used in combination with the contour integral representation. The surface energies on separate sides of the brane contain pole and finite contributions. Analytic expressions for both these contributions are derived. For an infinitely thin brane in odd spatial dimensions, the pole parts cancel and the total surface energy, evaluated as the sum of the energies on separate sides, is finite. For a minimally coupled scalar field the surface energy–momentum tensor corresponds to the source of the cosmological constant type.

A contour integral representation for the dual five-point function and a symmetry of the genus-4 surface in

The invention of the "dual resonance model" N-point functions BN motivated the development of current string theory. The simplest of these models, the four-point function B4, is the classical Euler Beta function. Many standard methods of complex analysis in a single variable have been applied to elucidate the properties of the Euler Beta function, leading, for example, to analytic continuation formulas such as the contour-integral representation obtained by Pochhammer in 1890. Here we explore the geometry underlying the dual five-point function B5, the simplest generalization of the Euler Beta function. Analyzing the B5 integrand leads to a polyhedral structure for the five-crosscap surface, embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120 in PGL(6). We find a Pochhammer-like representation for B5 that is a contour integral along a surface of genus five. The symmetric embedding of the five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the surface of genus four in R6 that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in O(6). The methods appear generalizable to all N, and the resulting structures seem to be related to associahedra in arbitrary dimensions.

On permanental polynomials of certain random matrices

The paper addresses the calculation of correlation functions of permanental polynomials of matrices with random entries. By exploiting a convenient contour integral representation of the matrix permanent some explicit results are provided for several random matrix ensembles. When compared with the corresponding formulae for characteristic polynomials, our results show both striking similarities and interesting differences. Based on these findings, we conjecture the asymptotic forms of the density of permanental roots in the complex plane for Gaussian ensembles as well as for the Circular Unitary Ensemble of large matrix dimension.

Asymptotic corrections to the eigenvalue density of the GUE and LUE

We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N×N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk.

Surface vacuum energy and stresses on a plate uniformly accelerated through the Fulling–Rindler vacuum

The vacuum expectation value of the surface energy–momentum tensor is evaluated for a massless scalar field obeying a mixed boundary condition on an infinite plate moving by uniform proper acceleration through the Fulling–Rindler vacuum. The generalized zeta function technique is used, in combination with the contour integral representation. The surface energies for separate regions on the left and on the right of the plate contain pole and finite contributions. Analytic expressions for both these contributions are derived. For a minimally coupled scalar, the surface energy–momentum tensor induced by vacuum quantum effects corresponds to a source of the cosmological constant type located on the plate and with the cosmological constant determined by the proper acceleration of the plate.

Surface Casimir densities and induced cosmological constant on parallel branes in AdS spacetime

Vacuum expectation value of the surface energy-momentum tensor is evaluated for a massive scalar field with general curvature coupling parameter subject to Robin boundary conditions on two parallel branes located on $(D+1)$-dimensional anti-de Sitter bulk. The general case of different Robin coefficients on separate branes is considered. As a regularization procedure the generalized zeta function technique is used, in combination with contour integral representations. The surface energies on the branes are presented in the form of the sums of single brane and second brane-induced parts. For the geometry of a single brane both regions, on the left (L-region) and on the right (R-region), of the brane are considered. The surface densities for separate L- and R-regions contain pole and finite contributions. For an infinitely thin brane taking these regions together, in odd spatial dimensions the pole parts cancel and the total surface energy is finite. The parts in the surface densities generated by the presence of the second brane are finite for all nonzero values of the interbrane separation. It is shown that for large distances between the branes the induced surface densities give rise to an exponentially suppressed cosmological constant on the brane. In the Randall-Sundrum braneworld model, for the interbrane distances solving the hierarchy problem between the gravitational and electroweak mass scales, the cosmological constant generated on the visible brane is of the right order of magnitude with the value suggested by the cosmological observations.

Casimir energy in the Fulling-Rindler vacuum

The Casimir energy is evaluated for massless scalar fields under Dirichlet or Neumann boundary conditions, and for the electromagnetic field with perfect conductor boundary conditions on one and two infinite parallel plates moving by uniform proper acceleration through the Fulling--Rindler vacuum in an arbitrary number of spacetime dimension. For the geometry of a single plate the both regions of the right Rindler wedge, (i) on the right (RR region) and (ii) on the left (RL region) of the plate are considered. The zeta function technique is used, in combination with contour integral representations. The Casimir energies for separate RR and RL regions contain pole and finite contributions. For an infinitely thin plate taking RR and RL regions together, in odd spatial dimensions the pole parts cancel and the Casimir energy for the whole Rindler wedge is finite. In $d=3$ spatial dimensions the total Casimir energy for a single plate is negative for Dirichlet scalar and positive for Neumann scalar and the electromagnetic field. The total Casimir energy for two plates geometry is presented in the form of a sum of the Casimir energies for separate plates plus an additional interference term. The latter is negative for all values of the plates separation for both Dirichlet and Neumann scalars, and for the electromagnetic field.

Nonsingular boundary integral method for deformable drops in viscous flows

A three-dimensional boundary integral method for deformable drops in viscous flows at low Reynolds numbers is presented. The method is based on a new nonsingular contour-integral representation of the single and double layers of the free-space Green’s function. The contour integration overcomes the main difficulty with boundary-integral calculations: the singularities of the kernels. It also improves the accuracy of the calculations as well as the numerical stability. A new element of the presented method is also a higher-order interface approximation, which improves the accuracy of the interface-to-interface distance calculations and in this way makes simulations of polydispersed foam dynamics possible. Moreover, a multiple time-step integration scheme, which improves the numerical stability and thus the performance of the method, is introduced. To demonstrate the advantages of the method presented here, a number of challenging flow problems is considered: drop deformation and breakup at high viscosity ratios for zero and finite surface tension; drop-to-drop interaction in close approach, including film formation and its drainage; and formation of a foam drop and its deformation in simple shear flow, including all structural and dynamic elements of polydispersed foams.

Closed form and infinite series solutions for the mgf of a dual-diversity selection combiner output in bivariate nakagami fading

Using a circular contour integral representation for the generalized Marcum-Q function, Q/sub m/(a,b), we derive a new closed-form formula for the moment generating function (MGF) of the output signal power of a dual-diversity selection combiner (SC) in bivariate (correlated) Nakagami-m fading with positive integer fading severity index. This result involves only elementary functions and holds for any value of the ratio a/b in Q/sub m/(a,b). As an aside, we show that previous integral representations for Q/sub m/(a,b) can be obtained from a contour integral and also derive a new, single finite-range integral representation for Q/sub m/(a,b). A new infinite series expression for the MGF with arbitrary m is also derived. These MGFs can be readily used to unify the evaluation of average error performance of the dual-branch SC for coherent, differentially coherent, and noncoherent communications systems.

Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram

The Schur process is a time-dependent analog of the Schur measure on partitions studied by A. Okounkov inInfinite wedge and random partitions, Selecta Math., New Ser.7(2001), 57–81. Our first result is that the correlation functions of the Schur process are determinants with a kernel that has a nice contour integral representation in terms of the parameters of the process. This general result is then applied to a particular specialization of the Schur process, namely to random 3-dimensional Young diagrams. The local geometry of a large random 3-dimensional diagram is described in terms of a determinantal point process on a 2-dimensional lattice with the incomplete beta function kernel (which generalizes the discrete sine kernel). A brief discussion of the universality of this answer concludes the paper.

Non-singular boundary-integral method for deformable drops in viscous flows

A three-dimensional boundary integral method for deformable drops in viscous flows at low Reynolds numbers is presented. The method is based on a new nonsingular contour-integral representation of the single and double layers of the free-space Green’s function. The contour integration overcomes the main difficulty with boundary-integral calculations: the singularities of the kernels. It also improves the accuracy of the calculations as well as the numerical stability. A new element of the presented method is also a higher-order interface approximation, which improves the accuracy of the interface-to-interface distance calculations and in this way makes simulations of polydispersed foam dynamics possible. Moreover, a multiple time-step integration scheme, which improves the numerical stability and thus the performance of the method, is introduced. To demonstrate the advantages of the method presented here, a number of challenging flow problems is considered: drop deformation and breakup at high viscosity ratios for zero and finite surface tension; drop-to-drop interaction in close approach, including film formation and its drainage; and formation of a foam drop and its deformation in simple shear flow, including all structural and dynamic elements of polydispersed foams. © 2004 American Institute of Physics. @DOI: 10.1063/1.1648639#

Magnetoelectric Green's functions and their application to the inclusion and inhomogeneity problems

Explicit expressions of magnetoelectric Green's functions are obtained for a transversely isotropic medium exhibiting coupling between the static electric and magnetic fields utilizing the contour integral representation. Four Green's functions exist which represent the coupled static electric and magnetic response to a unit point electric or magnetic charge. The Green's functions are applied to analyze the inclusion and inhomogeneity problems in an infinite magnetoelectric medium, and explicit, closed form expressions are obtained for the Eshelby type tensors. The magnetoelectric Eshelby's tensors can be readily used in the solution of numerous problems in the mechanics and physics of magnetoelectric solids.

Uniform asymptotic expansion of $J_\nu(\nu a)$ via a difference equation

There are two ways of deriving the asymptotic expansion of $J_\nu(\nu a)$ , as $\nu \to \infty$ , which holds uniformly for $a\geq 0$ . One way starts with the Bessel equation and makes use of the turning point theory for second-order differential equations, and the other is based on a contour integral representation and applies the theory of two coalescing saddle points. In this paper, we shall derive the same result by using the three term recurrence relation $J_{\nu+1}(x)+J_{\nu-1}(x)=(2\nu /x)J_\nu(x)$ . Our approach will lead to a satisfactory development of a turning point theory for second-order linear difference equations.

Computing Complex Airy Functions by Numerical Quadrature

Integral representations are considered of solutions of the Airy differential equation w′′−zw=0 for computing Airy functions for complex values of z. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss–Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.

Response function technique for calculating the random-phase approximation correlation energy

We develop a scheme to exactly evaluate the correlation energy in the random-phase approximation, based on linear response theory [Y. R. Shimizu, J. D. Garrett, R. A. Broglia, M. Gallardo, and E. Vigezzi, Rev. Mod. Phys. 61, 131 (1989)]. It is demonstrated that our formula is equivalent to a contour integral representation recently proposed [F. Donau, D. Almehed, and R. G. Nazmitdinov, Phys. Rev. Lett. 83, 280 (1999)] being numerically more efficient for realistic calculations. Examples are presented for pairing correlations in rapidly rotating nuclei.

Exact constructions of square-root Helmholtz operator symbols: The focusing quadratic profile

Operator symbols play a pivotal role in both the exact, well-posed, one-way reformulation of solving the (elliptic) Helmholtz equation and the construction of the generalized Bremmer coupling series. The inverse square-root and square-root Helmholtz operator symbols are the initial quantities of interest in both formulations, in addition to providing the theoretical framework for the development and implementation of the “parabolic equation” (PE) method in wave propagation modeling. Exact, standard (left) and Weyl symbol constructions are presented for both the inverse square-root and square-root Helmholtz operators in the case of the focusing quadratic profile in one transverse spatial dimension, extending (and, ultimately, unifying) the previously published corresponding results for the defocusing quadratic case [J. Math. Phys. 33, 1887–1914 (1992)]. Both (i) spectral (modal) summation representations and (ii) contour-integral representations, exploiting the underlying periodicity of the associated, quantum mechanical, harmonic oscillator problem, are derived, and, ultimately, related through the propagating and nonpropagating contributions to the operator symbol. High- and low-frequency, asymptotic operator symbol expansions are given along with the exact symbol representations for the corresponding operator rational approximations which provide the basis for the practical computational realization of the PE method. Moreover, while the focusing quadratic profile is, in some respects, nonphysical, the corresponding Helmholtz operator symbols, nevertheless, establish canonical symbol features for more general profiles containing locally-quadratic wells.

Note on classical string dynamics onAdS3

We consider bosonic strings propagating on Euclidean adS_3, and study in particular the realization of various worldsheet symmetries. We give a proper definition for the Brown-Henneaux asymptotic target space symmetry, when acting on the string action, and derive the Giveon-Kutasov-Seiberg worldsheet contour integral representation simply by using Noether's theorem. We show that making identifications in the target space is equivalent to the insertion of an (exponentiated) graviton vertex operator carrying the corresponding charge. Finally, we point out an interesting relation between 3D gravity and the dynamics of the worldsheet on adS_3. Both theories are described by an SL(2,C)/SU(2) WZW model, and we prove that the reduction conditions determined on one hand by worldsheet diffeomorphism invariance, and on the other by the Brown-Henneaux boundary conditions, are the same.

A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature

We treat the time discretization of an initial-value problem for a homogeneous abstract parabolic equation by first using a representation of the solution as an integral along the boundary of a sector in the right half of the complex plane, then transforming this into a real integral on the finite interval , and finally applying a standard quadrature formula to this integral. The method requires the solution of a finite set of elliptic problems with complex coefficients, which are independent and may therefore be done in parallel. The method is combined with spatial discretization by finite elements.

On Polynomials Related with Hermite–Padé Approximations to the Exponential Function

We investigate the polynomialsPn,Qm, andRs, having degreesn,m, ands, respectively, withPnmonic, that solve the approximation problem [formula]We give a connection between the coefficients of each of the polynomialsPn,Qm, andRsand certain hypergeometric functions, which leads to a simple expression forQmin the casen=s. The approximate location of the zeros ofQm, whenn⪢mandnequals;s, are deduced from the zeros of the classical Hermite polynomial. Contour integral representations ofPn,Qm,Rs, andEnmsare given and, using saddle point methods, we derive the exact asymptotics ofPn,Qm, andRsasn,m, andstend to infinity through certain ray sequences. We also discuss aspects of the more complicated uniform asymptotic methods for obtaining insight into the zero distribution of the polynomials, and we give an example showing the zeros of the polynomialsPn,Qm, andRsfor the casen=s=40,m=45.