Contour Integral Representation Research Articles

The spectral ζ-function for quasi-regular Sturm–Liouville operators

In this work, we analyze the spectral ζ-function associated with the self-adjoint extensions, TA,B, of quasi-regular Sturm–Liouville operators that are bounded from below. By utilizing the Green’s function formalism, we find the characteristic function, which implicitly provides the eigenvalues associated with a given self-adjoint extension TA,B. The characteristic function is then employed to construct a contour integral representation for the spectral ζ-function of TA,B. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the ζ-function to a larger region of the complex plane. We also present a method for computing the value of the spectral ζ-function of TA,B at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral ζ-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of s.

Reciprocal Hyperbolic Series of Ramanujan Type

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z=F12(1/2,1/2;1;x) and z′=dz/dx. When a certain parameter in these series is equal to π, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

A recursion relation for the number of Goldbach partitions of an even integer

The contour integral representation of the number of Goldbach partitions of an even integer, G(n), is extended to an integral with a support function that equals a linear combination of integers {G(m)}. A support function is found such that there is a nontrivial integral relation relating number of Goldbach partitions of n and m < n. The proof of the existence of a partition of any even integer greater than or equal to four into the sum of two primes follows from a recursion relation, resulting from an integral identity, that yields a non-zero lower bound for G(n). A partition of n, given a partition of n/2 , is derived from an algorithm based on solutions to congruence relations related to the roots of unity. This result provides a method for generating a prime partition of every even integer greater than or equal to 6 in addition to the above demonstration of its existence.

Heavy-traffic single-server queues and the transform method

Heavy-traffic limit theory is concerned with queues that operate close to criticality and face severe queueing times. Let W denote the steady-state waiting time in the GI/G/1 queue. Kingman (1961) showed that W, when appropriately scaled, converges in distribution to an exponential random variable as the system’s load approaches 1. The original proof of this famous result uses the transform method. Starting from the Laplace transform of the pdf of W (Pollaczek’s contour integral representation), Kingman showed convergence of transforms and hence weak convergence of the involved random variables. We apply and extend this transform method to obtain convergence of moments with error assessment. We also demonstrate how the transform method can be applied to so-called nearly deterministic queues in a Kingman-type and a Gaussian heavy-traffic regime. We demonstrate numerically the accuracy of the various heavy-traffic approximations.

The elliptic Ginibre ensemble: A unifying approach to local and global statistics for higher dimensions

The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows us to interpolate between the rotationally invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field at inverse temperature β = 2. Furthermore, it represents a determinantal point process in the complex plane with the corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici [Int. J. Mod. Phys. A 11, 941 (1996)]. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a known correspondence between non-interacting fermions in a trap in d real dimensions Rd and the d-dimensional harmonic oscillator. We present a rigorous proof for the local d-dimensional bulk (sine) and edge (Airy) kernel first defined by Dean et al. [Europhys. Lett. 112, 60001 (2015)], complementing the recent results by Deleporte and Lambert [arXiv:2109.02121 (2021)]. Using the same relation to the d-dimensional harmonic oscillator in d complex dimensions Cd, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in d real and d complex dimensions. For Cd with d = 1, this corresponds to non-interacting fermions in a rotating trap.

Prabhakar function of Le Roy type: a set of results in the complex plane

Two families of Le Roy-type functions are considered in this paper, respectively with 3 and 4 indices. In the case with 4 indices, these special functions called to be as Prabhakar analogues, are proved to be entire functions and their order and type are determined. Some Mellin–Barnes-type contour integral representations as well as the Mellin transforms are derived. It is also established that the nth derivatives of the 3-parametric Le Roy-type functions are Le Roy-type functions with 4 indices (of Prabhakar type). This result is further used for representing the 3-index function in a Taylor series. Analogical relations for the integrals and derivatives of fractional orders are also obtained. Finally, some interesting particular cases of the discussed special functions are considered.

The novel approach to the closed-form average bit error rate calculation for the Nakagami-m fading channel

The research presents a procedure of the closed-form average bit error rate evaluation for wireless communication systems in the presence of multipath fading. A generalization of the classical moment generating function is applied, and a connection between the Hankel-type contour-integral representation of the Marcum Q-function and Gauss Q-function is obtained and applied to the analytic derivation of the quadrature amplitude modulated signal average bit error rate in the presence of Nakagami-m fading. The correctness of the obtained solution was verified, and its computational efficiency (in terms of accuracy and time gain), compared with the prevailing approximation, was demonstrated. The proposed methodology can be extended to a wide variety of composite channels.

Mathematical model of a vertical-axial wind motor in a viscous gas flow

The development of vertical axial wind turbines in Ukraine is in its infancy for many reasons: the lack of systematic theoretical and experimental studies of the aerodynamic characteristics of various schemes of wind turbines, the lack of an appropriate experimental base in technical universities, design organizations, insufficient number of available publications in foreign literature due to high competition between by monopoly firms. At present, various numerical methods are widely used to solve urgent problems of aero hydrodynamics, which are used for the approximate solution of boundary value problems in the form of differential forms of mathematical models. Their common disadvantages are the particularity and laboriousness of solutions, high requirements for computing resources, and, as a consequence, the complexity of solving optimization problems and economic feasibility. These problems can be avoided by using exact or approximate analytical dependences, which allow solving some urgent problems of studying the interaction of a viscous gas with the bearing elements of both aircraft and engineering structures. The existing methods for calculating the aerodynamic characteristics, based on the ideology of the mathematical model of the motion of an ideal medium without viscous interaction, do not correspond to the real processes and demands of practice. The article presents the ideology of determining the aerodynamic characteristics of the interacting system of solid profiles in the configuration of a vertical-axial wind turbine in a viscous gas flow. Based on generalized vector-tensor analysis, contour integral representations of solutions to the main problem of fluid and gas mechanics related to the determination of kinematic and dynamic characteristics of interaction have been constructed. In addition, the existence of a vector potential of the tensor of stresses and deformation velocities has been proved, reducing, in the simplest cases, the process of determining characteristics to integration. The limit values of these integral representations are a system of boundary integral equations, allow for elementary algorithmization, and lead to a system of linear algebraic equations having a single solution.

Integral Representation of Polynomial Xn (x;a,b)

Abstract: In the present paper we have obtained fine difference formula, contour integral representation, real integral representation, infinite single integral representation, finite single integral representation, finite double integral representation, finite double integral representation of polynomial ࢔ࢄ .(࢈ ,ࢇ ;࢞) Keywords: Finite difference, single integral representation, contour integral representation, simple generating relation, double integral representation

Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface

Abstract We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel–Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus $0$, then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.

On certain results related to the hypergeometric function FK

Motivated by some recent applications of a certain triple series hypergeometric function in the areas of physics and statistics, we investigate further Saran's work (e.g., [Acta Math. 93 (1955), 293–312]) related to this useful hypergeometric function FK. Its analytic continuation and asymptotics are obtained by establishing a new single contour integral representation. With the help of the fractional integration by parts, we prove a new integral identity and discuss in detail its various special cases and their relationship with Koschmieder's work [Acta Math. 79 (1947), 241–254]. Several consequences and special cases of the results presented in this paper are also mentioned.

On A Logarithmic Mittag-Leffler Function, its Properties and Applications

In this paper, we introduce a logarithmic Mittag-Leffler function and discuss some of its properties. The application of these properties become helpful in extension of Pochhammer’s type contour integral representations and Rodrigues formulae of some known hypergeometric functions. On application point of view, some relations are discussed which are useful in interpreting the phenomenon of spread of infectious diseases in terms of Lauricella’s multiple hypergeometric functions.

An efficient high-frequency method of the EM near-field scattering from an electrically large target

Abstract In this paper, an efficient high-frequency method combining physical optics and shooting and bouncing rays is proposed to calculate the electromagnetic near-field scattering from the electrically large target (such as ship) in the near zone. In order to solve the mono-static and bi-static scattering problems, the local expansion technique based on the facets of target is presented to account for the near-field scattering, which has a singularity-free characteristic. And then the near-field contour-integral representation is introduced by refining the formulations in traditional far-field calculation to reduce the computational complexity. Therefore, this method is more straightforward to deal with the near-field scattering problem. Finally, some analyses on the near-field scattering characteristics and Doppler spectra of a surveillance ship in near zone are investigated by using the proposed method.

Generalizations and applications of Srinivasa Ramanujan’s integral associated with infinite Fourier sine transforms in terms of Meijer’s G-function

Abstract In this paper, we obtain analytical solutions of an unsolved integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} of Srinivasa Ramanujan [S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 1915, 75–86] with suitable convergence conditions in terms of Meijer’s G-function of one variable, by using Mellin–Barnes type contour integral representations of the sine function, Laplace transform method and some algebraic properties of Pochhammer’s symbol. Also, we have given some generalizations of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} in the form of integrals ℧ S * ⁢ ( υ , b , c , λ , y ) {\mho_{S}^{*}(\upsilon,b,c,\lambda,y)} , Ξ S ⁢ ( υ , b , c , λ , y ) {\Xi_{S}(\upsilon,b,c,\lambda,y)} , ∇ S ⁡ ( υ , b , c , λ , y ) {\nabla_{S}(\upsilon,b,c,\lambda,y)} and ℧ S ⁢ ( υ , b , λ , y ) {\mho_{S}(\upsilon,b,\lambda,y)} with suitable convergence conditions and solved them in terms of Meijer’s G-functions. Moreover, as applications of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} , the three new infinite summation formulas associated with Meijer’s G-function are obtained.

Spherical Spin Glass Model with External Field

We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We compute the limiting values and fluctuations of the free energy as well as three types of overlaps in the setting where the strength of the external field goes to zero as the dimension of the spin variable grows. In particular, we consider overlaps with the external field, the ground state, and a replica. Our methods involve a contour integral representation of the partition function along with random matrix techniques. We also provide computations for the matching between different scaling regimes. Finally, we discuss the implications of our results for susceptibility and for the geometry of the Gibbs measure. Some of the findings of this paper are confirmed rigorously by Landon and Sosoe in their recent paper which came out independently and simultaneously.

Numerical evaluation of Mittag-Leffler functions

The Mittag-Leffler function is computed via a quadrature approximation of a contour integral representation. We compare results for parabolic and hyperbolic contours, and give special attention to evaluation on the real line. The main point of difference with respect to similar approaches from the literature is the way that poles in the integrand are handled. Rational approximation of the Mittag-Leffler function on the negative real axis is also discussed.

A Class of Functions satisfying certain boundary value problems

In this paper for constructing of a class of functions consisting of integral representations, we introduce a double integral, a formula pertaining to extended fractal strings, consisting of separate variables functions in the integrand. Further by this double integral formula, we determine various functions, integrals and contour integral representations on introducing different special functions for these integrand functions connecting to RiemannLiouville and Weyl fractional integral functions. Finally, we apply our obtained results to find certain boundary value problems and present precise examples.

Matrix representations for a certain class of combinatorial numbers associated with Bernstein basis functions and cyclic derangements and their probabilistic and asymptotic analyses

In this paper, we mainly concerned with an alternate form of the generating functions for a certain class of combinatorial numbers and polynomials. We give matrix representations for these numbers and polynomials with their applications. We also derive various identities such as Rodrigues-type formula, recurrence relation and derivative formula for the aforementioned combinatorial numbers. Besides, we present some plots of the generating functions for these numbers. Furthermore, we give relationships of these combinatorial numbers and polynomials with not only Bernstein basis functions, but the two-variable Hermite polynomials and the number of cyclic derangements. We also present some applications of these relationships. By applying Laplace transform and Mellin transform respectively to the aforementioned functions, we give not only an infinite series representation, but also an interpolation function of these combinatorial numbers. We also provide a contour integral representation of these combinatorial numbers. In addition, we construct exponential generating functions for a new family of numbers arising from the linear combination of the numbers of cyclic derangements in the wreath product of the finite cyclic group and the symmetric group of permutations of a set. Finally, we analyse the aforementioned functions in probabilistic and asymptotic manners, and we give some of their relationships with not only the Laplace distribution, but also the standard normal distribution. Then, we provide an asymptotic power series representation of the aforementioned exponential generating functions.

Rate of convergence for products of independent non-Hermitian random matrices

We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in terms of a uniform Kolmogorov-like distance. First, we prove that for products of Ginibre matrices, the optimal rate is given by O(1∕n), which is attained with overwhelming probability up to a logarithmic correction. Avoiding the edge, the rate of convergence of the mean empirical spectral distribution is even faster. Second, we show that also products of matrices with independent entries attain this optimal rate in the bulk up to a logarithmic factor. In the case of Ginibre matrices, we apply a saddlepoint approximation to a double contour integral representation of the density and in the case of matrices with independent entries we make use of techniques from local laws.

The Generalized MGF Approach to Closed-Form Average Symbol Error Rate Calculation

The research presents the generalization of the moment generating function approach to the problem of the closed-form average symbol error rate calculation (ASER). A Hankel-type contour integral representation of the Gauss Q-function is obtained based on its connection with the generalized Marcum Q-function. The closed-form solutions for the integrals of the averaged Gaussian Q-function and its squared version were derived for the factorized power-type moment generating function (MGF). The results were applied to the analytic derivation of the quadrature amplitude modulated (QAM) signal ASER and its asymptotic form in presence of Fluctuating Beckmann fading.