TRIGONOMETRIC APPROXIMATIONS FOR SOME BESSEL FUNCTIONS

Formulas are obtained for approximating the tabulated Bessel functions Jn(x), n = 0–9 in terms of trigonometric functions. These formulas can be easily integrated and differentiated and are convenient for personal computers and pocket calculators.


INTRODUCTION
Ordinary Bessel functions of the first kind Jn(x) arise in many physical and engineering applications.These functions must often be approximated by appropriate formulas suitable for implementation using personal computers and pocket calculators.Although main-frame computer routines are available for evaluating Bessel functions they are not suitable for implementation using pocket calculators and personal computers.In an attempt to obtain formulas for approximate com- putation of Bessel functions, Waldron [1] obtained trigonometric ap- proximations for Jo(x) and Jl(X) from which Jn(x), n > can be obtained using the recurrence relation Jn-1 (x) + Jn+l (X) --2n Jn(x) X (1) However, (1) can yield accurate results only if n < x, otherwise severe accumulation of rounding errors will result [2].Moreover, the factor Using (2), ( 5)-(13), the Bessel functions Jn(X), n 0-9 were calculated and the results are shown in Tables I-X together with the corre- sponding values obtained from mathematical tables [2].From columns and 4 of Tables I-X one can easily see that the accuracy of the  [2] Eq. ( 14) o(x)   Jl(X) Jl(X) Tables [2] Eq. ( 14) eq. ( 5)    J4(x) Tables [2] gq. ( 14      approximations of ( 5)-( 13) deteriorates as the order, n, of the Bessel function Jn(x) increases and also as the argument, x, increases.
The major intention of this paper is, therefore, to present new trigonometric approximate expressions for the Bessel functions of any order n.These expressions are obtained by approximating the tabu- lated values of Bessel functions using Fourier-series representation.
Comparisons with tabulated values of Bessel functions show that the proposed approximations enjoy excellent accuracies.

PROPOSED APPROXIMATIONS
Here we propose to approximate the Bessel function Jn(x), in the range x < B, using the Fourier-series of (14).

Jn(X)
anmCOS --X (14) m=O In general, the parameters a0 and anm can be obtained using standard curve-fitting or discrete Fourier transfer (DFT) techniques.These techniques invariably demand extensive computing facilities and well- developed software.Alternatively, by using short-cut methods [4, 5], the parameters anm can be obtained without recourse to the DFT or standard curve-fitting routines.However, these methods result in Fourier-series having only a finite number of terms.This raises the question of the order of the Fourier-series as this will significantly affect the accuracy of the resulting approximations.In general, a high-order Fourier-series requires a proportionally large number of data points input to the short-cut methods as well as the DFT and the standard curve-fitting techniques.However, using the procedure described in [6] it is possible, at least in theory, to obtain an infinite-order Fourier- series model using a finite number of data points.Thus the number of terms of the Fourier-series can be arbitrarily increased until any desired degree of accuracy is achieved.In principle the procedure described in [6] is based on a short-cut approach in which the function J,(x) is mirror-imaged and then interpolated between the tabulated data points by using straight line segments joined end-to-end as shown in Figure 1.Using the slopes of the straight-line segments of Figure 1, it is easy ; -,.-" slope slope One Nil period 2 B FIGURE Tabulated data after mirror imaging and approximation by straight line segments.
using the procedure described by Kreyszig [7] to show that the parameters anm can be expressed by -B mr anm (mTr)2 o1 OK-I -if--(Ok+l Cek) COS Xk+l (15) k=l The results obtained for the Bessel functions Jn(x), n 0-9, with B 17.5 for Jo(x)-Jl(X) and B 20 for J3(x)-J9(x), are shown in Tables XI and XII.
Using the parameters of Tables XI and XII and Eq. ( 14) calcula- tions were made and the results are shown in Tables I-X.Comparing columns 2 and 3 of Tables I-X one can easily see that the proposed approximations for the Bessel functions J,,(x), n 0-9 are in excellent agreement with their tabulated values.

CONCLUSION
In this paper new formulas for approximating Bessel functions Jn(x), n 0-9 have been presented.These formulas, in terms of trigono- metric functions, can be easily integrated and differentiated and are, therefore, convenient for further mathematical processing.Although the approximations presented here are valid for values of x < 20, their extension to cover wider ranges of arguments x is straight forward.The proposed approximations do not use any recurrence relation and, therefore, avoid the severe accumulation of rounding errors.Finally, it is worth mentioning that the algorithm used for obtaining the present approximations for the Bessel functions J,,(x) is general and can be used for obtaining trigonometric approximations for any tabulated functions.

TABLE V
Comparison of J4(x) with its approximations J4(x)

TABLE XI Parameters
anm of the Bessel functionJn(x)

TABLE XII
Parameters a,,m of the Bessel function J,,(x)