Evaluation of the Modified Bessel Function of the First Kind and Zeroth Order

We investigate the internal space of Bessel functions which is associated to thegroup Z of positive and negative integers defining their orders. As a result wepropose and prove a new unifying formula (to be added to the huge literatureon Bessel functions) generating Bessel functions of real orders out of integerorder ones. The unifying formula is expected to be of great use in appliedmathematics. Some applications of the formula are given for illustration

: The report contains tables of the first five roots of the following transcendental equations: (a) J0(alpha) Y0(alpha) = Y0(alpha) J0(alpha); (b) J1(alpha) Y1(alpha) = Y1(alpha) J1(alpha); (c) J0(alpha) Y1(alpha) = Y0(alpha) J1(alpha) where J0(alpha), Y0(alpha), J1(alpha), Y1(alpha) are Bessel functions of order 0 and 1 respectively. In these equations, alpha is the unknown and k is a parameter which may assume any positive value, other than 0 or 1. Additional tables are included listing an auxiliary quantity gamma which is better suited to interpolation particularly when k is close to unity. (Author)

In recent days, relay assisted cellular networks are gaining more importance in research and development because of the recent adoption of new communication standards with relaying and cooperation communication. This has introduced a multichannel diversity along with the multiuser diversity and the channel aware dynamic resource allocation models. The issue of the optimal location of relays has risen especially when dedicated relays are used as the standard proposes instead of the cooperative model of the users. In this paper, we study the optimal location of a single relay. Furthermore, we study the effect of changing the number of users on the optimal location of the relay. The effect of adding multiple relays to the system is examined. The optimal locations are examined when the relay channels are the only channels to be used by the system and when the direct channel (DT) is also available. The problem formulation assumes, Rayleigh block faded channels, half duplex regenerative (repetition coding) decode-and-forward (DF) relaying strategy, long-term average total transmitted power constraint and orthogonal multiplexing of the users messages within the channel blocks. New simple and accurate approximations to the modified Bessel functions of the first kind, zeroth order I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> (z) and first order I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> (z) are presented. The new proposed approximations are given as a simple finite sum of scaled exponential functions. Comparisons are made between the exact functions, classic approximations, and the new approximation in terms of simplicity and accuracy. The new approximation proves to be sufficiently accurate to bridge the gap between the classic large and small argument approximations and has potential applications in allowing one to analytically evaluate integrals containing Modified Bessel Functions, yielding simple closed-form solutions. A generalized closed-form expression for the average bit error rate over Nakagami-n (Rice) fading, and Rayleigh fading as a special case, are derived as sample applications, and the results are compared with Monte Carlo Simulation, where a very good matching is achieved.

Applications of a simple approximation of Bessel functions of integer order, in terms of trigonometric functions, are discussed for several examples from electromagnetism and optics. The method may be applied in the intermediate regime, bridging the ‘small values regime’ and the ‘asymptotic’ one, and covering, in this way, an area of great physical interest. The examples that illustrate our approach are accessible to the undergraduate student.

Let \(I_0\) and \(K_0\) be modified Bessel functions of the zeroth order. We use Vanhove’s differential operators for Feynman integrals to derive upper bounds for dimensions of the \({\mathbb {Q}}\)-vector space spanned by certain sequences of Bessel moments $$\begin{aligned} \left\{ \left. \int _0^\infty [I_0(t)]^a[K_0(t)]^b t^{2k+1}\mathrm {d}t\right| k\in {\mathbb {Z}}_{\ge 0}\right\} , \end{aligned}$$where a and b are fixed non-negative integers. For \(a\in {\mathbb {Z}}\cap [1,b)\), our upper bound for the \( {\mathbb {Q}}\)-linear dimension is \(\lfloor (a+b-1)/2\rfloor \), which improves the Borwein–Salvy bound \(\lfloor (a+b+1)/2\rfloor \). Our new upper bound \(\lfloor (a+b-1)/2\rfloor \) is not sharp for \(a=2, b=6\), due to an exceptional \({\mathbb {Q}}\)-linear relation \(\int _0^\infty [I_0(t)]^2[K_0(t)]^6 t\mathrm {d}t=72\int _0^\infty [I_0(t)]^2[K_0(t)]^6 t^{3}\mathrm {d}t\), which is provable by integrating modular forms.

This paper aims to shape the identification of thin inhomogeneities with different dielectric/magnetic properties from a two-dimensional homogeneous background. The shapes are identified through subspace migration without requiring the diagonal elements of the collected multi-static response matrix. To understand why subspace migration without diagonal elements can retrieve the shape of a thin inhomogeneity, we carefully investigate the relations between the imaging function and Bessel functions of orders 0 and 1. This analysis exploits the fact that when a thin homogeneity exists, the measured far-field pattern can be represented as an asymptotic expansion formula. The investigated relation is supported in numerical experiments with noisy data.

TRAITEMENT AUTOMATIQUE DES SONDAGES ELECTRIQUES AUTOMATIC PROCESSING OF ELECTRICAL SOUNDINGS*

The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real quadratic field K and even integer k > 2 and construct a series of functions co,,(Zl, z2) (m=0, 1, 2, . . .) which are modular forms of weight k for the Hilbert modular group SL2(9 ((9=ring of integers in K). The form co o is a multiple of the Hecke-Eisenstein series for K, while all of the other co,. are cusp forms. The Fourier expansion of co,. (z 1, z2) is calculated in w 2; each Fourier coefficient is expressed as an infinite sum whose typical term is the product of a finite exponential sum (analogous to a Kloosterman sum) and a Bessel function of order k 1 . The main result is that, for any points z 1 and z 2 in the upper half-plane .~, the numbers m k-1 co,,(z 1, z2) (m= 1, 2, ...) are the Fourier coefficients of a modular form (in one variable) of weight k. More precisely, let D be the discriminant of K, e.=(D/ ) the character of K, and S(D, k, ~) the space of cusp forms of weight k for Fo(D ) with character e; then for fixed z 1, zzc .~, the function

An investigation is made of the differential equations d 2w 1 da; ■ l, /t2—1 f,&gt;4 d ? = i d ^ + r + V - + / ( w ) r } in which u is a large complex parameter, u a real or complex parameter independent of u , and z is a complex variable whose domain of variation may depend on arg u and u , and need not be bounded. General conditions are obtained under which solutions exist having the formal series w oo A P(Z) 5=0“ + f'(z) u2 V s=0«2* as their asymptotic expansions for large | u|, uniformly valid with respect to z, arg u and u. Here P(z) is respectively an exponential function, Airy function or Bessel function of order u , and the coefficients As and B5 are given by recurrence relations.

From Mie's classical theory of the action of small spherical particles on plane waves of light, the expression giving the loss of light due to absorption and scattering is reduced to the formula involving only Bessel functions of orders given by half integral values. The result is used for calculating the absorption by small carbon particles whose diameter is comparable with the wave-length of the incident light, particles that can be measured only by interference methods. When the diameter is less than 0.2 μ the coefficient of absorption decreases toward the red end of the spectrum. The reverse is true for 0.3 and 0.4 μ particles.

We have found new exact solutions of the coupled nonlinear Schroedinger equation of Benney-Roskes which is written as i u t -β u x x +γ u y y +δ u * u u -2 w u =0, β w x x +γ w y y -βδ( u * u ) x x =0. Three types of new solutions have been obtained. They are described by the Hermite polynomials, the Bessel function of order ±1/4 and ±1/3 (Airy function). Physically these solutions represent exploding and decaying solitons.

We derive a semiclassical expression for an energy smoothed autocorrelation function defined on a group of eigenstates of the Schr\"odinger equation. The system we considered is an energy-conserved Hamiltonian system possessing time-invariant symmetry. The energy smoothed autocorrelation function is expressed as a sum of three terms. The first one is analogous to Berry's conjecture, which is a Bessel function of the zeroth order. The second and the third terms are trace formulae made from special trajectories. The second term is found to be direction dependent in the case of spacing averaging, which agrees qualitatively with previous numerical observations in high-lying eigenstates of a chaotic billiard.

Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function${K}_{ir} (x)$of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of${K}_{ir} (x)$and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of$r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of${K}_{ir} (x)$.

CHAPTER 17 - Bessel Functions

The Neumann expansion of Bessel functions (of integer order) of a function \(g:\mathbb {C}\rightarrow \mathbb {C}\) corresponds to representing g as a linear combination of basis functions φ0, φ1, …, i.e., \(g(s)=\sum _{\ell = 0}^\infty w_\ell \varphi _\ell (s)\), where φi(s) = Ji(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.