A new polynomial approximation for Jm Bessel

Applied Mathematics and Computation 183 (2006) 1220–1225 www.elsevier.com/locate/amc

A new polynomial approximation for Jm Bessel functions L-L. Li a, F. Li a

a,*

, F.B. Gross

b

Institute of Electronics, EM Theory and Application, Chinese Academy of Sciences, 19 Beisihuan, Beijing 100080, China b Advanced Technology, Argon ST, 12701 Fair Lakes Circles, USA

Abstract New approximate solutions to the m-order Bessel function of the first kind are derived. The solution is expressed as a polynomial for Bessel function of all fractional order polynomial. Comparisons are made between the exact solution and the new approximation. One utility of this derivation is to allow researchers to analytically evaluate integrals containing Bessel functions. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Polynomial approximation; m-order Bessel functions; Modified Bessel functions; Weber functions; Neumann functions; Hankel functions

1. Introduction The m-order Bessel function appeared frequently in numerous problem solutions such as EM scattering, elasticity, fluid flow, acoustics, and communications [1–5]. The Bessel function of the first kind is typically expressed as an infinite series using the method of Frobenius or in integral form using the Poisson integral. There is, however, no exact closed form expression available. Therefore, in each application one must resort to numerical integration techniques, finite difference methods, and series or polynomial approximations in order to calculate numerical values. Although accurate numerical approximations are available and used in most software applications, there is an advantage to expressing m-order Bessel functions in the form of polynomial approximations. Some existing intractable problems involving Bessel functions might be reduced to closed form approximations using these new polynomial expansions. An accurate closed form approximation can provide quicker computer simulations because convergence is no longer an issue. Gross [2,5] has derived a simple polynomial approximation to J0 and J1, and Millance and Eads [6] obtained the integer order polynomial using the derivatives relationship. This paper extends those approximation methods to include all m-order Bessel functions.

*

Corresponding author. E-mail address: fl[email protected] (F. Li).

0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.06.047

L-L. Li et al. / Applied Mathematics and Computation 183 (2006) 1220–1225

1221

2. Polynomial approximation to m-order Bessel function The m-order Bessel function of the first kind can be expressed in integral form [6] as Z 1 x m 1 pffiffiffi J m ðxÞ ¼ ð1 x2 Þm1=2 cosðxxÞdx: 2 pCðm þ 1=2Þ 1

ð1Þ

An integral was introduced by Gross when studying the general problem of EM scattering from a conducting strip grating [1]. It is given as Z 2 d cosðxÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð2nxÞdx; ð2Þ f2n ðkÞ ¼ p 0 2 k sin2 ðxÞ where k = sin d and 0 6 d 6 p/2. Gross derived a polynomial approximation to Eq. (2) which led to an approximation to the 0th order Bessel function. We can generalize Eq. (2) to include the order m such that, Z 2 d cosðxÞ cosð2nxÞdx; ð3Þ f2n ðk; mÞ ¼ p 0 k 2 sin2 ðxÞ1=2m where m is an arbitrary real number and now Eq. (3) bears a resemblance to Eq. (1). Eq. (3) can also be expanded into a polynomial expression leading to an approximation to Eq. (1). This can be shown by the following development. Since k = sin(d) for small arguments (d ! 0) we can allow k ! d. We can also make a change of variable, such that x = xd. Thus we can recast Eq. (3) as Z 2 1 cosðxdÞ 2m f2n ðd; mÞ ð4Þ 1=2m cosð2nxdÞd dx: p 0 sin2 ðxdÞ 1 d2 Furthermore, by defining d = x/2n and letting d ! 0 we have x 2 x 2m Z 1 cosðxxÞ ;m f2n dx: 1=2m 2n p 2n 0 ð1 x2 Þ It can be seen that Eq. (5) bears a striking resemblance to Eq. (1). That is m pffiffiffi 2 p J m ðxÞ n2v f2n ðx=2n; vÞ: x Cðm þ 1=2Þ

ð5Þ

ð6Þ

A polynomial expression for Eq. (3) can be derived in a similar fashion as was performed in [5]. Again, allowing sin x = k sin a, and after several manipulations one can reduce Eq. (3) to the form Z p=2 2 f2n ðk; mÞ ¼ k 2m cos2m ðaÞ cosð2n arcsinðk sinðaÞÞÞda: ð7Þ p 0 Since arcsin(k sin a) = p/2 arccos(k sin a), cos(np + /) = (1)ncos(/), and using the definition of a Chebyshev polynomial one can derive the following expression: Z p=2 2 n f2n ðk; mÞ ¼ k 2m ð1Þ cos2m ðaÞT 2n ðk sin aÞda ð8Þ p 0 with T 2n ðk sin aÞ ¼ Chebyshev polynomial of order 2n: Using Gradsthteyn and Ryzhik [7], the Chebyshev polynomial, of order 2n, can be described by n X m ð2n m 1Þ! 2n2m ð2xÞ ð1Þ : T 2n ðxÞ ¼ n m!ð2n 2mÞ! m¼0 Substituting Eq. (9) into Eq. (8) one can derive an alternative expression.

ð9Þ

1222


f2n

" # n 2n2mþ2m Z p=2 X 2 n m ð2n m 1Þ! 2n2m x 2n2m 2m ; m ¼ ð1Þ n 2 ð1Þ cos ðaÞ sin ðaÞda : 2n p m!ð2n 2mÞ! 2n 0 m¼0

x

The integral in Eq. (10) can be solved using the Beta function such that Z p=2 cos2m ðaÞ sin2ðnmÞ ðaÞda ¼ Bðm þ 1=2; n m þ 1=2Þ;

ð10Þ

ð11Þ

0

where Bðm þ 1=2; n m þ 1=2Þ ¼

1 Cðm þ 1=2ÞCðn m þ 1=2Þ : 2 Cðm þ n m þ 1Þ

From Eqs. (6) and (10) we have a polynomial approximation for the m-order Bessel function given as " # n x m n X x 2n2m ð2n m 1Þ! Cðn m þ 1=2Þ mþn pffiffiffi J^m ðxÞ ¼ ð1Þ ; 2 m!ð2n 2mÞ! Cðm þ n m þ 1Þ n p m¼0

ð12Þ

where J^m ðxÞ ¼ finite series approximation to J m ðxÞ: The approximation Eq. (12) is only valid for the range 0 < x < 2n because assumptions were made for the argument x/2n < 1 in Eq. (5). The accuracy increases as x/2n ! 0. Thus the fractional-order polynomials given in Eq. (12) are sufficient to approximate Jm(x). Since 2n > x then the number of series terms in Eq. (12) must be an integer, such that n > int(x/2). As n increases, the polynomial accuracy increases. For small Bessel function arguments, a low order polynomial approximation is adequate. After some algebraic manipulation we can recast Eq. (12) in the form n x2mþm X ^ J^m ðxÞ ¼ bm;n;m 2 m¼0 for 0 < x < 2n; ð13Þ n X ^m ðxÞ ¼ r m¼0

where ðn þ m 1Þ! n12m ^ ; bm;n;m ¼ ð1Þm m!ðn mÞ! Cðm þ m þ 1Þ x 2mþm ^m ðxÞ ¼ ^ bm;n;m : r 2 This formulation looks similar to the classic Bessel function infinite series expansion using the method of Taylor which is given in [7] as 1 1 x 2mþm X X bm;n;m ¼ rm ðxÞ; ð14Þ J m ðxÞ ¼ 2 m¼0 m¼0 where ð1Þm ; m!Cðm þ m þ 1Þ x 2mþm : rm ðxÞ ¼ bm;n;m 2 bm;n;m ¼

The series coefficients for the new series approximation, ^bm;n;m , differ from the Taylor series coefficients by three terms. Thus, ðn þ m 1Þ! 12m ^ n bm;n;m : bm;n;m ¼ ðn mÞ!

ð15Þ


1223

From this expression it can be easily proven that when n ! 1 for any given m ^bm;n;m ¼ bm;n;m . In the case the new solution is degenerated into the Taylor series. 3. Discussion and numerical results To determine the efficacy of the new series approximation Eq. (13), we can compare it to the truncated Taylor series Eq. (14) and also to the exact solution. For n = 10, m = 0, 1.5, 3, we can generate a family of m-order Bessel function plots as shown in Fig. 1. The truncated Taylor series becomes unstable and diverges for increasing x values whereas the new series solution remains finite but with increasing inaccuracy. The new series solution provides a good approximation provided that x < n. In order to demonstrate the divergence of our solution, we perform the Cauchy ratio test [8] applied to both series of Eq. (13). The Cauchy ratio test is a measure of the convergence of a series by comparing adjacent increasing series terms. The Cauchy ratio test for the new series in Eq. (13) is given as r ^mþ1 ðxÞ ^m ðxÞ ¼ : ð16Þ q ^ ðxÞ r m

After some manipulations it can be obtained that h x i2 m 2 1 ^m ðxÞ ¼ 1 q : n ðm þ 1Þðm þ m þ 1Þ 2

ð17Þ

^m ðxÞ ! 0, that is the new solution is absolutely It shows that when m ! 1, for any x and any m the ration q convergent. It is also instructive to look at the percentage error between the new solution and the exact solution. Fig. 2 indicates the percentage error for n = 20, and m = 0, 1.5, 3.

Fig. 1. Comparison of Jm curves.

1224


Fig. 2. Percentage error between exact Jm(x) and new solution J^m ðxÞ.

It should be noted that the error decreases as n increases regardless of the order of the Bessel function. Fig. 3 shows a plot of the decreasing error vs. increasing n for the 3rd order Bessel function with x = 12. In general less than 4% error can be maintained provided that x < 0.6n. Because the form of the new series solution is identical to the form of the Taylor series, the same Bessel function rules apply to derivatives and recurrence relationships. Eq. (13) satisfies the iterative relationship for Bessel functions. That is 2m J^mþ1 ðxÞ ¼ J^m ðxÞ J^m1 ðxÞ: x

ð18Þ

Also, since our series solution is identical in form to the Taylor series, then we have also derived an approximation to the modified Bessel function of the first kind and m-order. Thus,

Fig. 3. Decreasing error with increasing n.


Î m ðxÞ ¼ jm J^m ðjxÞ ¼

n X

jm ^ bm;n;m

m¼0

2mþm jx : 2

1225

ð19Þ

Any linear combinations of Bessel functions can also be approximated by this new series derivation. Therefore, the finite series in Eq. (13) can be utilized to define an approximate solution for all Bessel functions including Bessel functions of the second kind (Weber or Neumann functions) J^m ðxÞ cosðmpÞ J^m ðxÞ : Y^ m ðxÞ ¼ sinðmpÞ

ð20Þ

Hankel functions of the first kind ^ m ðxÞ ¼ J^m ðxÞ þ jY^ m ðxÞ H

ð21Þ

and Hankel functions of the second kind b m ðxÞ ¼ J^m ðxÞ j Yb m ðxÞ: H

ð22Þ

Finally, because the new series approximation is of the same form as the classic infinite series approximation, the new solution is also valid for complex arguments. 4. Conclusion A fractional order polynomial is derived which accurately approximates the m-order Bessel function. The polynomial solution is cast in a truncated series form where the series coefficients are similar to the Taylor series coefficients; however, they converge more quickly. Unlike the truncated Taylor solution, the new series does not diverge for large x. In a similar way, the new derivation can be used to approximate modified Bessel functions Im(x), Bessel functions of the second kind Ym(x), and Bessel functions of the third kind H mð1Þ ðxÞ and H mð2Þ ðxÞ. The accuracy of the new series solution exponentially improves by increasing the number of series terms. The utility of this new series solution is the ability to approximate and simplify existing problems involving Bessel function solutions, speed the computation of Bessel function values, and find closed form approximate solutions from existing open form problems. Acknowledgements This work was supported in part by National Natural Science Foundation of China under Grant No. 10375071 and Foundation of the National Key Lab. of Electromagnetic Environment under Grant No. 51486020203 ZK1301. References [1] F.B. Gross, W. Brown, New frequency-dependent edge mode current density approximations for TM scattering from a conducting strip grating, IEEE Trans. Antennas. Propagat. 41 (9) (1993) 1302–1307. [2] F.B. Gross, in: Bessel Functions, in: John G. Webster (Ed.), Wiley Encyclopedia of Electrical and Electronics Engineering, vol. 2, John Wiley Interscience, 1999. [3] C.F. Du Toit, The numerical computational of Bessel functions of the first and second kind for integer orders and complex arguments, IEEE Trans. Antennas. Propagat. 38 (9) (1990) 1341–1349. [4] G. Matriyenko, On the evaluation of Bessel functions, Appl. Comput. Harmon. Anal. 1 (1) (1993) 116–135. [5] F.B. Gross, New approximations to J0 and J1 Bessel function, IEEE Trans. Antenna Propagat. 43 (August) (1995) 904–907. [6] R.P. Millance, J.L. Eads, Polynomial approximation to Bessel function, IEEE Trans. Antenna Propagat. 51 (June) (2003) 1398–1400. [7] I.S. Gradsthteyn, I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1980, p. 959. [8] H. Cohen, Mathematics for Scientists and Engineers, Prentice-Hall, 1992, p. 128.