A General Analytic Approximation Technique for the Modified Bessel Functions with Fractional Order Iv (x), Applied to I1/6(x) and I1/7(x)

A general technique to obtain simple analytic approximations for the first kind of modified Bessel functions. The general procedure is shown, and the parameter determination is explained through the applications to this particular case I1/6(x)and I1/7(x). In this way, it shows how to apply the technique to any particular orderν, in order to obtain an approximation valid for any positive value of the variable x. In the present method power series and asymptotic expansion are simultaneously used. The technique is an extension of the multipoint quasirational approximation method, MPQA. The main idea is to look for a bridge function between the power and asymptotic expansion of the I1/6(x), and similar procedure for I1/7(x). To perform this, rational functions are combined with hyperbolic ones and fractional powers. The number of parameters to be determined for each case is four. The maximum relative errors are 0.0049 for ν=1/6, and 0.0047 for ν=7. However, these relative errors decrease outside of the small region of the variables, wherein the maximum relative errors are reached. There is a clear advantage of this procedure compared with any other ones.

Duality between Bessel Functions and Chebyshev Polynomials in Expansions of Functions

In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.

THIRD-ORDER DIFFERENTIAL SUBORDINATION RESULTS FOR ANALYTIC FUNCTIONS INVOLVING THE GENERALIZED BESSEL FUNCTIONS

In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel functions of the first kind, which is defined as z（Bκ＋1^c f（z））′= κBκ^c f（z）-（κ- 1）Bκ＋1^c f（z）,where b, c, p ∈ C and κ = p ＋（b ＋ 1）/2 ∈ C ／ Z0^-（Z0^-= {0,-1,-2, … }）. The results are obtained by considering suitable classes of admissible functions. Various known or new special cases of our main results are also pointed out.

Measurement of Optical Fiber Geometry Parameters with Bessel Function Fitting Method

Geometry parameters of optical fiber are crucial in evaluating the quality of the optical fiber.Near⁃field light distribution method is recommended in GB15972.20-2008 for the measurement of geometry parameters.To distinguish the boundary between fiber core and cladding,it is necessary to illuminate the fiber.The end face of the core is a bright spot with unclear edge,so the true edge of the core and the cladding cannot be accurately judged.A method is proposed in this paper to measure the geometry parameters of optical fiber by Bessel function fitting.Theoretically,the solution to the electromagnetic vector of mode field satisfies Bessel function,and the boundary between the core and the cladding can be precisely extracted by Bessel function fitting.Edges of the fiber were fitted by elliptical curves,and the geometry parameters of the fibers could be calculated.Results show that the maximum deviations of the diameters and the average differences of the fibers were decreased under normal and abnormal conditions respectively.The proposed method is an efficient way to obtain edge data and can improve the accuracy and stability of geometry parameters of optical fibers.

SIMPLIFICATION OF FREQUENCY EQUATION OF MULTILAYERED CYLINDERS AND SOME RECURSION FORMULAE OF BESSEL FUNCTIONS

In this paper the axially symmetric, radial vibration frequency equation of multilayered cylinders made of the same materials in plain strain state is studied. It is proved in this paper that the frequency equation of:several contact hallow cylinders can be replaced by the frequency equation of a single hollow cylinder, so that the solving process is greatly simplified. Some recursion formulae of Bessel functions are derived from such practical problems as well.

Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind

In the paper,by virtue of a general formula for any derivative of the ratio of two differentiable functions,with the aid of a recursive property of the Hessenberg determinants,the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.

CHEBYSHEV APPROXIMATION OF THE SECOND KIND OF MODIFIED BESSEL FUNCTION OF ORDER ZERO

The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational Chebyshev basis.

Study Hankel Transforms and Properties of Bessel Function via Entangled State Representation Transformation in Quantum Mechanics

In Phys. Lett. A 313 （2003） 343 we have found that the self-recipràcal Hankel transformation （HT） is embodied in quantum mechanics by a transform between two entangled state representations of continuum variables. In this work we study Hankel transforms and properties of Bessel function via entangled state representations＇ transformation in quantum mechanics.

Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros

The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.

Certain Integral Transforms of Generalized k-Bessel Function

The objective of this note is to provide some(potentially useful) integral transforms(for example, Euler, Laplace, Whittaker etc.) associated with the generalized k-Bessel function defined by Saiful and Nisar [3]. We have also discussed some other transforms as special cases of our main results.

On <i>q</i>-Analogues of Laplace Type Integral Transforms of <i>q</i>²-Bessel Functions

The present paper deals with the evaluation of the q-Analogues of Laplece transforms of a product of basic analogues of q2-special functions. We apply these transforms to three families of q-Bessel functions. Several special cases have been deducted.

Multiplication and Translation Operators on the Fock Spaces for the q-Modified Bessel Function

We study the multiplication operator M by z2 and the q-Bessel operator Δq,αon a Hilbert spaces Fq,α of entire functions on the disk D( o, ) , 0qq,α into itself. Next, we study a generalized translation operators on Fq,α .

Bessel Function and Damped Simple Harmonic Motion

A glance at Bessel functions shows they behave similar to the damped sinusoidal function. In this paper two physical examples (pendulum and spring-mass system with linearly increasing length and mass respectively) have been used as evidence for this observation. It is shown in this paper how Bessel functions can be approximated by the damped sinusoidal function. The numerical method that is introduced works very well in adiabatic condition (slow change) or in small time (independent variable) intervals. The results are also compared with the Lagrange polynomial.

An integral estimate of Bessel function and its application

In this paper the authors give a new integral estimate of the Bessel function, which is an extension of Calderón-Zygmund’s result. As an application of this result, we prove that the parameterized Marcinkiewicz integral μ Ω ρ with variable kernels is of type (2, 2), where the kernel function Θ does not have any smoothness on the unit sphere in ? n .

Error Bounds for Uniform Asymptotic Expansions-Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe?cients and the remainders in the Taylor expansions of analytic functions.By using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.

ON THE ZEROS OF QUOTIENTS OF BESSEL FUNCTIONS

The paper studies the zeros of 2mIm/Im-1- (m+1)I1 /I0, where Im are the Bessel functions.

Elastodynamic Infinite Elements with Modified Bessel Shape Functions for Dynamic Soil-Structure Interaction

The paper is devoted to formulations of decay and mapped elastodynamic infinite elements, based on modified Bessel shape functions. These elements are appropriate for Soil-Structure Interaction （SSI） problems, solved in time or frequency domain and can be treated as a new form of the recently proposed Elastodynamic Infinite Elements with United Shape Functions （EIEUSF） infinite elements. The formulation of 2D Horizontal type Infinite Elements （HIE） is demonstrated here, but by similar techniques 2D Vertical （VIE） and 2D Comer （CIE） Infinite Elements can also be formulated. Using elastodynamic infinite elements is the easier and appropriate way to achieve an adequate simulation including basic aspects of Soil-Structure Interaction. Continuity along the artificial boundary （the line between finite and infinite elements） is discussed as well and the application of the proposed elastodynamic infinite elements in the Finite Element Method （FEM） is explained in brief. Finally, a numerical example shows the computational efficiency of the proposed infinite elements.

基于格林函数法研究分数阶Bessel-Gaussian光束近场传输

为揭示FBG(fractional-order Bessel-Gaussian)光束在近场传输时的光强和相位演化情况,基于光传播的独立性和叠加性原理,通过引入虚光源点,并结合格林函数法,严格推导了FBG光束的非傍轴积分表达式,并进一步利用非傍轴下的各阶修正解,精确计算了FBG光束轴上的光强分布.计算结果表明:对于分数阶Bessel-Gaussian光束的远场传输,利用经典的傍轴近似理论可以得到准确的计算结果,但在近场传输时,傍轴近似理论的计算结果误差很大,只有利用非傍轴修正解才能精确研究FBG光束的近场光学传输.研究结果为将FBG光束更好地应用于近场光学奠定必要的理论基础.

On the Total Dynamic Response of Soil-Structure Interaction System in Time Domain Using Elastodynamic Infinite Elements with Scaling Modified Bessel Shape Functions

This paper is devoted to a new approach—the dynamic response of Soil-Structure System (SSS), the far field of which is discretized by decay or mapped elastodynamic infinite elements, based on scaling modified Bessel shape functions are to be calculated. These elements are appropriate for Soil-Structure Interaction problems, solved in time or frequency domain and can be treated as a new form of the recently proposed elastodynamic infinite elements with united shape functions (EIEUSF) infinite elements. Here the time domain form of the equations of motion is demonstrated and used in the numerical example. In the paper only the formulation of 2D horizontal type infinite elements (HIE) is used, but by similar techniques 2D vertical (VIE) and 2D corner (CIE) infinite elements can also be added. Continuity along the artificial boundary (the line between finite and infinite elements) is discussed as well and the application of the proposed elastodynamical infinite elements in the Finite element method is explained in brief. A numerical example shows the computational efficiency and accuracy of the proposed infinite elements, based on scaling Bessel shape functions.

Some Generalized q-Bessel Type Wavelets and Associated Transforms

In this paper wavelet functions are introduced in the context of q-theory. We precisely extend the case of Bessel and q-Bessel wavelets to the generalized q-Bessel wavelets. Starting from the （q,v）-extension （v = （α,β）） of the q-case, associated generalized q-wavelets and generalized q-wavelet transforms are developed for the new context. Reconstruction and Placherel type formulas are proved.