Introducing A New Method to Obtain Analytic Approximations of the Bessel Function 𝐽1(x) |Chapter 7 | Physical Science: New Insights and Developments Vol. 3

 

The power series of 𝐽1(x) is well known and its convergence radius is infinite. In this work, an analytic approximation for 𝐽1(x) has been found, which is simple and precise, and good for most of the applications of these functions in Physics. Two techniques have been used here, and the simplest approximant is a function of four parameters. The technique used here resembles a little the Pade method, since rational functions are used, but now this type of function is combined in an efficient way with elementary functions. Furthermore, series power and asymptotic expansions are used simultaneously, as in the Multipoint Quasi-rational Approximation MPQA method. However, here important improvements have been introduced. Though the form of the approximate is built considering the above two expansions, however the parameters of the approximations are determinates by two methods, one similar to the minimum square error method and the other using the coefficients of two expansions, power and asymptotic. The resulting approximations are very simple yet achieve high accuracy, sufficient for most physical applications of 𝐽1

 

Author(s) Details

Pablo Martin
Department of Physics, Universidad de Antofagasta, Av. Angamos 601, Antofagasta, Chile.

 

Fernando Maass
Department of Physics, Universidad de Antofagasta, Av. Angamos 601, Antofagasta, Chile.

 

Please see the book here :- https://doi.org/10.9734/bpi/psniad/v3/6605

Potential of High Accuracy Approximation for the Modified Bessel Function of Fractional Order \(I_{1/3}(x)\), Using MPQA Method with Hyperbolic Functions | Chapter 7 | Current Research Progress in Physical Science Vol. 2

The modified Bessel functions of fractional order \(I_{1/3}(x)\) has been approximated by an analytic function containing rational and hyperbolic functions. The Bessel functions the order 1/3 are very important, because its connection with the Airy functions. A technique using both power series and assymptotic expansion has been used. An approximation has been found for the modified Bessel function \(I_{1/3}(x)\). The accuracy of the approximation is very high using only three parameters., and the largest relative error is smaller than 0,004.

 

Author(s) Details

P. Martin

Department of Physics, Universidad de Antofagasta, Antofagasta, Chile.

Jorge Olivaresl

Department of Mathematics. Universidad de Antofagasta, Antofagasta, Chile.

E. Valero

Department of Mathematics Career, Universidad Mayor de San Andrés, La Paz, Bolivia.

 

Please see the link:- https://doi.org/10.9734/bpi/crpps/v2/516

 

Bounded Linearness of Distributional Generalized Hankel-Schwartz Type Transformations on L’ p,v Spaces | Chapter 12 | Current Topics on Mathematics and Computer Science Vol. 7

 This research defines two Hankel-Schwartz transformations. The Hankel-Schwartz type transformations are specified on the Lp,v and (L' p,v) spaces. The transformations defined by (1) and (2) are also shown to be bounded linear operators of Lp,v. On the Lp,v spaces, we also investigate the behaviour of transformations defined by (3) and (4). It is further demonstrated that the transformations specified by (3) and (4) are bounded linear operators of Lp,v into Lp,2p(3(alpha)+(beta))-v and Lp,-v-2((alpha)-(beta))p, respectively. Finally, we demonstrated that distributional generalised HankelSchwartz type transformations on (L' p,v) spaces are linearly limited.

Author (S) Details

B. B. Waphare
MAEERs MIT Arts, Commerce and Science College Alandi (D), Pune, Maharashtra, India.

View Book :- https://stm.bookpi.org/CTMCS-V7/article/view/2941