Struve Functions Research Articles

The spherical Bessel and Struve functions and operational methods

We review some aspects of the theory of spherical Bessel functions and Struve functions by means of an operational procedure essentially of umbral nature, capable of providing the straightforward evaluation of their definite integrals and of successive derivatives. The method we propose allows indeed the formal reduction of these family of functions to elementary ones of Gaussian type. We study the problem in general terms and present a formalism capable of providing a unifying point of view including Anger and Weber functions too. The link to the multi-index Bessel functions is also briefly discussed.

Infinite integrals involving Bessel functions by an improved approach of contour integration and the residue theorem

Two lemmas are established to evaluate integrals involving Bessel functions or products of two Bessel functions on a large semicircle. These lemmas are as useful as Jordan’s lemma. With the help of them, the approach of contour integration and residue theorem to infinite integrals becomes much more powerful than ever before, yet is still simple and easy. A general formula is derived for infinite integrals involving rational functions, powers and Bessel functions, the index of the power and the order of the Bessel function being independent, and the result is expressed in terms of residues and coefficients of the Laurent series of the rational function. Similar formulae are presented for infinite integrals involving Neumann or Struve functions. Another general formula is derived for infinite integrals involving rational functions, powers and two Bessel functions, the index of the power and the orders of the Bessel functions being independent. The formulation presented may be extended to infinite integrals of similar types. Typical examples are worked out for each type of integral, and many of the results obtained seem not available in the literature.

Geometric Properties of Generalized Struve Functions

Geometric Properties of Generalized Struve Functions

Determination of the Electromagnetic Field Created by Line Current and Sheet Current Source at the Earth’s Surface

The electromagnetic field that generated by line current and sheet current at the surface of the earth can be expressed in analytical form. The line current created at the earth’s surface by an infinitely long line current is given by the inverse Fourier integrals over a horizontal wave number. The sheet current can be obtained by integrating the line current expansions using a Neumann and Struve functions; these functions have known mathematical properties, including the series expansions. The series expansions are exact with neglecting the displacement currents. Assuming a uniform earth and that there is no propagation, the three nonzero field components can be expressed in terms of the Neumann and Struve functions. The integrals of line current expansions are calculated by using the numerical methods. The results represented graphically and illustrated by figures. Results can be used to evaluate numerical solutions of more complicated modeling algorithms.

Uniformly Starlike Functions and Uniformly Convex Functions Associated with the Struve Function

In this paper, our goal is to determine sufficient conditions for the family of Struve functions in order to belong to the classes of uniformly convex functions and uniformly starlike functions in the open unit disk U. Several corollaries and consequences of our main results are also derived.

Functional inequalities for modified Struve functions II

In this paper our aim is to prove some monotonicity and convexity results for the modified Struve function of the second kind by using its integral representation. Moreover, as consequences of these results, we present some functional inequalities (like Tur\'an type inequalities) as well as lower and upper bounds for modified Struve function of the second kind and its logarithmic derivative.

The low frequency axisymmetric modal sound radiation efficiency of an elastically supported annular plate

Abstract This paper deals with the sound radiation efficiency of a vibrating, thin, elastically supported annular plate embedded into a flat rigid baffle. The free axisymmetric time harmonic vibrations have been considered for a single mode. It has been assumed that the influence of the air column above the plate on the plate's vibrations is negligible. First, the sound radiation efficiency has been formulated as an integral. Further, rigorous mathematical manipulations have been carried out based on the theory of summation of multiple expansion series containing the hypergeometric functions. As a result, the formulations have been expressed as some fast convergent expansion series containing only the Bessel and Struve functions of integer order and the spherical Bessel functions. The presented formulations of sound radiation efficiency of an elastically supported annular plate are useful for numerical calculations within the low frequency range what is important for practical reasons. The formulations are valid for axisymmetric boundary conditions and they enable changing the values of boundary stiffness constants. Consequently, the analysis of influence of the plate's edge attachment on the sound radiation efficiency has been performed. The limiting transitions have also been performed from formulations valid for the elastically supported annular plates to the formulations valid for annular plates with classical boundary conditions (clamped, simply supported and free) at one edge or at both edges.

Infinite integrals involving Bessel functions by contour integration

The approach of contour integration and residue theorem is systematically developed to evaluate infinite integrals involving Bessel functions or closely related ones. Three independent formulations applicable to different types of integrals are presented. The conventional formulation using Hankel functions as auxiliary functions, whose previous development leave considerable room for improvement, is fully developed. The second and third formulations, which appear unnoticed before, uses as auxiliary functions linear combinations of Bessel and Struve functions, and linear combinations of Anger and Weber functions, respectively. These functions play a crucial role as the Hankel functions do in the conventional formulation. Several general formulae are derived, each expressing an infinite integral in terms of residues. The integrands are products of rational functions, powers and Bessel (Neumann, Struve, Anger or Weber) functions, with some relation satisfied between the index of the power and the order of the Bessel or related function. Typical examples are worked out for each type of integral.

Magnetoelastic Field of a Multilayered and Functionally Graded Cylinder With a Dynamic Polynomial Eigenstrain

In this paper, an analytical solution is obtained for the magnetoelastic response of a multilayered and functionally graded cylinder with an embedded dynamic polynomial eigenstrain. The internal core of the cylinder endures a harmonic eigenstrain of cubic polynomial distribution along the radial direction. Both plane strain and plane stress conditions are assumed for the axisymmetric cylinder. The composite cylinder is placed in a constant magnetic field parallel to its axis. The magnetoelastic governing equations are solved exactly and the displacement and stress components are obtained in terms of Bessel, Struve, and Lommel functions. Using the analytical solution for the multilayered, composite cylinder, the magnetoelastic response of a functionally graded cylinder with exponential and power law distribution of material properties is investigated. Finally, the numerical results reveal the effects of external magnetic field, eigenstrain, and nonhomogeneity indices on the magnetoelastic response of the heterogeneous cylinders.

Hardy space of generalized Struve functions

In this paper, we present some immediate applications of convexity involving Struve functions associated with the Hardy space of analytic functions, i.e. we obtain conditions for the functionto belong to the Hardy space

Theoretical and numerical results on effects of attenuation on correlation functions of ambient seismic noise

S U M M A R Y We study analytically and numerically effects of attenuation on cross-correlation functions of ambient noise in a 2-D model with different attenuation constants between and outside a pair of stations. The attenuation is accounted for by quality factor Q(ω) and complex phase velocity. The analytical results are derived for isotropic far-field source distribution assuming the Fresnel approximation and mild attenuation. More general situations including cases with non-isotropic source distributions are examined with numerical simulations. The results show that homogeneous attenuation in the interstation regions produces symmetric amplitude decay of the causal and anticausal parts of the noise cross-correlation function. The attenuation between the receivers and far-field sources generates symmetric exponential amplitude decay and may also cause asymmetric reduction of the causal/anticausal parts that increases with frequency. This frequency dependence can be used to distinguish asymmetric amplitudes due to attenuation from frequency-independent asymmetry in noise correlations generated by nonisotropic source distribution. The attenuations both between and outside station pairs also produce phase shifts that could affect measurements of group and phase velocities. In terms of noise cross-spectra, the interstation attenuation is governed by Struve functions while the attenuation between the far-field sources and receivers is associatedwith exponential decay and the imaginary part of complex Bessel function. These results are fundamentally different from previous studies of attenuated coherency that append the Bessel function with an exponential decay that depends on the interstation distance.

Integral representations and summations of the modified Struve function

It is known that Struve function $\mathbf H_\nu$ and modified Struve function $\mathbf L_\nu$ are closely connected to Bessel function of the first kind $J_\nu$ and to modified Bessel function of the first kind $I_\nu$ and possess representations through higher transcendental functions like generalized hypergeometric ${}_1F_2$ and Meijer $G$ function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for $\mathbf L_\nu(x)$. In this paper firstly, we obtain various another type integral representation formulae for $\mathbf L_\nu(x)$ using the technique developed by D. Jankov and the authors. Secondly, we present some summation results for different kind of Neumann, Kapteyn and Schl\"omilch series built by $I_\nu(x)$ and $\mathbf L_\nu(x)$ which are connected by a Sonin--Gubler formula, and by the associated modified Struve differential equation. Finally, solving a Fredholm type convolutional integral equation of the first kind, Bromwich--Wagner line integral expressions are derived for the Bessel function of the first kind $J_\nu$ and for an associated generalized Schl\"omilch series.

Improvement of Self-Inductance Calculations for Circular Coils of Rectangular Cross Section

In this paper, an improvement for calculating the self-inductance of circular coils of rectangular cross section is presented. An integral of elementary functions is applied to replace the Bessel and Struve functions in the integrand of the inductance expression, and the term of improper integral without the exponential factor is solved to a closed form which consists of the complete elliptic integrals of the first and second kind and generalized hypergeometric functions. Due to this method, the main difficulties of the numerical evaluations of the inductance expression are overcome. Excellent agreement with published method is confirmed by the numerical comparisons and the calculations can be accelerated to several tens to hundreds of times depending on the shape factors of the coils. Similar expressions of the special cases (thin-wall solenoids and disk coils) are also given and compared with previous methods. Furthermore, the asymptotic behaviors of the special coils which tend to be a circular ring are obtained and numerically validated.

On a Unification of Generalized Mittag-Leffler Function and Family of Bessel Functions

In the present work, a unification of certain functions of mathematical physics is proposed and its properties are studied. The proposed function unifies Lommel function, Struve function, the Bessel-Maitland function and its generalization, Dotsenko function, generalized Mittag-Leffler function etc. The properties include absolute and uniform convergence, differential recurrence relation, integral representations in the form of Euler-Beta transform, Mellin-Barnes transform, Laplace transform and Whittaker transform. The special cases namely the generalized hypergeometric function, generalized Laguerre polynomial, Fox H-function etc. are also obtained.

On a treatment of coupled integral equations in nonlinear unilateral contact problem of plates

Application of finite Hankel integral transforms to the problem of a uniformly loaded unilaterally supported rectangular thin plates, the solution of partial differential equation can then be reduced to two coupled Fredholm integral equations of the second kind that obtained from the coupled dual-series equations of mixed boundary conditions, and also together with two constraint integral equations resulted from both of zero corner force condition and compatibility of edge displacements near each plate corner that loses contact. Owing to the summation of many different infinite series in the kernels and inhomogeneous parts of equations for the requirements of attainable highest accuracy and reducible computational time, the modified mathematical manipulations are improved and required asymptotic expansion for large arguments of the Bessel and Struve functions is treated to accelerate the convergence of series rapidly. The latter is to avoid the undesirable lengthy computer runs in numerical computations. Consequently, this paper is devoted to demonstrate the mathematical description and numerical algorithms in order to solve the system of coupled integral equations.

Starlikeness and Convexity of Generalized Struve Functions

We give sufficient conditions for the parameters of the normalized form of the generalized Struve functions to be convex and starlike in the open unit disk.

Homogenization of the elastic properties of pyrolytic carbon based on an image processing technique

AbstractIn this work, the linear elastic material properties of differently textured variants of pyrolytic carbon are homogenized from the submicro‐ to the micro‐scale. In high resolution transmission electron microscope (HRTEM) lattice fringe images, the microstructure of pyrolytic carbon manifests itself in terms of projections of graphene layers. According to their orientation distribution, different textures of pyrolytic carbon have been classified. Assuming a von Mises‐Fisher distribution for the spatial orientation of single graphene layers, the orientation distribution function of the projected layers in the image plane is analytically found to be a modified Struve function. For each pyrolytic carbon texture, Maximum‐likelihood estimates for the mean orientation and the concentration parameter of the von Mises‐Fisher distribution are obtained numerically. Hereby, Fourier transformation and appropriate filters are used to determine the probabilities for discrete orientations of the graphene layers directly from HRTEM images. First‐ and second‐order bounds of the linear elastic properties of pyrolytic carbon of the different textures are computed. Elastic constants of graphite and pyrolytic graphite have been used for modeling the elastic behavior of the graphene layers within a continuum mechanical setting. Due to the high anisotropy of all analyzed textures of pyrolytic carbon, the differences even between the second‐order bounds are quite large.

Dimensioning the Sine‐Generated Curve Meander Geometry1

Mecklenburg, Daniel E. and Anand D. Jayakaran, 2012. Dimensioning the Sine‐Generated Curve Meander Geometry. Journal of the American Water Resources Association (JAWRA) 48(3): 635‐642. DOI: 10.1111/j.1752‐1688.2012.00638.xAbstract: The sine‐generated curve (SGC) presented by Langbein and Leopold (1966) is widely acknowledged as an idealized description of river meander form. However, as originally presented, the SGC application to the field, laboratory, or models is not a trivial task. The SGC function simply provides the direction of the flow path as its rate of change continually varies from most gradual at the crossover to most rapid at the apex of bends. Although the commonly used dimensions of meander length, stream length per meander, sinuosity, and amplitude have been approximated previously, this technical note precisely defines these dimensions by solving the integrals of the sine and cosine of the SGC function. This is performed using integral representations of the Bessel and Struve functions for the zero order and their power‐series expansions. Higher precision offered by this technique will be of particular value to modeling and flume‐based studies where better conformity to the SGC is necessary. The note goes on to present two simple line and arc patterns optimally fit to the SGC as simplified yet more precise methods for practical design in river restoration schemes. Equations are presented defining the geometry of the arc and line pattern and relating meander length, amplitude, radius of curvature, and sinuosity.

Exact solutions for the mutual inductance of circular coils and elliptic coils

An exact solution is presented for the mutual inductance between general noncoaxial thin circular and elliptic coils with parallel axes. The thin coil solution is given as an angular integral of an elliptic integral expression. In addition, for the coaxial case, an exact solution is given for the mutual inductance of a thick circular coil and a thick elliptic coil. The elliptic coil is such that the coil thickness is the same along both elliptic semi-axes. The thick coil solution is given as an integral of an expression involving Bessel and Struve functions. Extensive numerical results for sample geometries are given for both solutions, which are cross checked against each other in the limit as the thick coils become thin.

Simple analytic formula for the period of the nonlinear pendulum via the Struve function: connection to acoustical impedance matching

An approximate formula for the period of pendulum motion beyond the small amplitude regime is obtained based on physical arguments. Two different schemes of different accuracy are developed: in the first less accurate scheme, emphasis is given on the non-quadratic form of the potential in connection to isochronism, and a specific form of a generic formula that is met in many previous works is produced, while the second and main result contains the Struve function which is further approximated by a simple sinusoidal expression based on its maximum value. The accuracy of the final formula gives a relative error of less than 0.2% for angles up to 140°. In addition, a simple relation between the Struve function and the complete elliptic integral of the first kind is produced, since they both constitute solutions of the pendulum period. This relation makes it possible for someone to connect different areas in physics and solve a difficult task by comparison with another much more simple one. As an example, a connection between the pendulum period and the acoustical radiation impedance is proposed through impedance matching and some interesting relations are produced. This paper is intended for undergraduate students to be useful for analysing pendulum experiments in introductory physics labs and it is also believed to offer valuable insight into some properties of the simple pendulum in undergraduate courses on general physics.