INVESTIGACiÓN REVISTA MEXICANA DE FíSICA 44 (1) 231-234 JUNIO I99R

A new wide-rauge approximation of modified Besscl functions iu terms ofelementary functions

D. Kh. MorozovInstituto de Ciencias Nucleares, Universidad Nacional Autónoma de México

Apartado postal 70-543. 04510 México. D. F, Mexico

V.V. VoilsekhovichInstituto de Astmnomia, Universidad Nacional Autónoma de México

Apartado posta/70-264, 04510 Mexico, D.F, Mexico

Recibido el 6 de enero de 1997; aceptado el 9 de febrero de 199:-:

A new approximation fm modified Bessel functions in terms of elementary functions is proposcd. In (ontrast with the existing approxima.tions.the express ion derivcd is valid for any positive values of arguments and orders al' modilled Besscl functions. The errors of approximationare investigated nurnerically. It is shown that the magnitudc of error of the approximation decreases rapidly with an increase of the arder.having a maximum fm Bessel function of arder lero. The results ohtained are compared witll tabulated ones and illustrated graphically. Awide rangc of validity and relatively small magnitude of errors makcs the approximation useful for applications.

Key",ords: Modificd Bessel functions; clementary functions

Se propone una nueva aproximación en términos dc funciones elementales para las funciones de Besscl modificadas. En contraste con lasaproximaciones existentes. la expresión derivada es válida para cualquier valor positivo de los argumentos. así como cualquier orden de lasfunciones de Bessel modificadas. La propagación del error se ha investigado numéricamcntc. Se dcmuestra que la magnitud del error de laaproximación decrece rápidamente con el incremento del orden. teniendo un máximo para la función dc Besscl de orden ceTO.Los resultadosobtenidos se comparan con los tabulados y se i1ustrnn gráficamente. El amplio rango dc validcz. y la relativamente pequeña magnitud de loserrorcs. hace esta aproximación útil para una gran variedad de aplicacioncs.

Descriptores: Funciones de Bessel modificadas; funciones elementales

PAes: 02.30.Gp: 02.30.Mv

1. Introduction On Ihe olher hand, Ihere is Ihe asymptotic expansion [IJ

(4/)' - 1) (4v" - 9)+ ,2! (8x)

which IS applicablc whcn v' » x.Another ascending series and asymplotic expansions as-

sociated with the modifled Bessel functions are known:asymptotic expansion for large arguments and fixed order,asymptotic expansion for large orders and fixed argumentoDcbyc's asymptotic expansions. uniform asymptotic cxpan-sioos, elc. (fm more information see Ref. 1). AH the abo verepresenlations are applicahle illside houndcd inlervals of :f

and v. On lhe other hand. Pade approximations were pro-posed 14,5J which allows (lne 10 approximate [,,(x) for anyvalue oflhe argulllenl and order. However, despite of the widcrange of validity. the lasl approximations are cumbersorne foranalytical calculations.

The aim ol' this paper is a presentation of lhe simpleand wide-ranged continuolls approximation for (he modifiedBessel functions convenient for analytical calculations. Such

+].(4v' - 1) (4/)' - 9) (4v' - 25)33' (8x)

They are rclalcd to the firSI-kind Besscl functionsJ,,(x) as IIJ

(X)" = (x2/4)'I,,(x)= 2' I:klr(v+k+l)'

k=O

where r denotes the Gamma-l'unction. This series convergesquickly while X « v [41.

[., d" d (2 2)].r--

2+ x-

1- x + v [.(x) = O.

dI (X

Thc functions 1/.1 (x) arc well-invesligatcd theorcli-cally 12] as well as Ihcre is a number ol' successful nUll1cricalalgorithms for thcir calculalion [3].

The Taylor's series expansion 01'[,,(1:) has Ihe form

Thc moditicd Bessel functions 1/.1 (x) are finding widc appli-cations in diffcrcnt areas 01' physics. Thesc functions salisfy(he following sccond-order ordinary differential cquation ¡1):

232 n.KII. MOROZOV ANJ) v.v. VOITSEKH()vICH

2. Approximate formula

\vhcrc I' dcnotes lhe Gamma function.LClllS approxilllale lhe integrand in Eq. (1) as follo\Vs:

13

579

10 12 14 16 18 20 22

X

(a)

8 10 12 14 16 18 20

X(b)

í7

, 2

s

'O -3~EW .5

•.~ .7;;-¿;a: .,

~é.E

"Exoji .1o.<l

022

• '"e0.18~

u 0.16e~ 0.14U.

-¿; 0.12•• 0.10•"" 0.08 I

,1! 0.06'Oo 0.04

" 0.02

0.00

2

( I )

(.1m"1,,(.1) = r ("+ 1/2) r (1/2)

r1 ') lJ-l/'2x./_Idl(l-t-) exp(:¡;t),

Thc following integral representation of 1//(.1') holds [G]:

"pproximalioll is useful rol' physical applicalions involvingthese fUllctions where analylical results are oC main interest:L'valualion of integrals, extreme analysis, differenlial equa-tiolls \Vith cocflicienls including modifieJ Bessel functions,clc.

Our allalysis is reslriclcd hy only real, non-negalivc val-lIC.'"nI' thc argument .1" and order //. Applying the inlegralrcpn:scntatinn ror mndifled Besscl fUllCtiol1s. we derive theIll'L,tkd expression rol' arhitrary x ami 11 » 1 without any fit-ling paramctcrs. Then we show Ihat the validity range or Ihisexprcssioll can he exlended for the arhitrary /1 lIsing somesmall corrcctions. In ordcr to show the errors 01' approxillla-¡ion \Ve prescnt the comparison nI' llulIlerical ami approximateca!Culalions uf 1//(.1').

Tllcn. c4ualing the magniludes 01' the inlegrand and ap~proximatc funclioll at 1 = 'o we grllhc following expressionfor ..l:

Choosing f" in such a \vay that the Illaximum of the ap-proximatc fUllction coincides with lhe maximum 01' the inte-grand, \Ve have

")"_1/' [(1 - to)'](1 - t- I'xp(,rl) '" Aexp rr'

,,o' ( JI + //' /r' - //(1')"1,,(.1') '" , " 1/4

1 (1' + 1/2) (.1'2 + //')

X .'xp ( -// + J,r' + //2) . (6)

Thc approximation formula can be simplifico making use01' lhe following asymptotic expansion of Ihe Gamma func-tion:

F[(iURE l. Approximnlion 01"modificd Bcsscl fUllctions (a) nndrclalivc crrors 01' approxim;llion (h) ror IJ = 1,3. G, /. n. ExplicitamI approxirnatc 1'unctiotls are shown hy solid nnd datted ¡ines.rcspcctivcly. Approximation givcn by Eq. (8) hns bcen uscd. Thcdigits inside lhe plo1 shm••..lhe orders of the functions.

(.1)

(2)

(21~-, 1)',

2.l"2// - 1

,0= ----+~.r

( ")"-1/'A = 1 - lií ex!, (:z:l.o). (4) [' (" + 1/2) '" .j2;;//" ex!, (-//). (7)

Finally, equating lhe second derivatives 01' the :ntegrandami approximatc I'unction at f = 'o \ve determine (1:

Substiluting (7) into (6) we get a rclatively simple approx-imation ror f,,(:r):

For Ihe case 01' 11 » 1 ¡he integration in Eq. (1) canhe pcrforlned in inlinite limils. Suhstiluting Eqs. (2)-(5) intoEq. (1) and performing Ihe intcgration. wc have the followingapproximation 01' f,,(;r) ror II » 1:

1 - 15,,.to +" + 1/2'

(5)

(JI + //' (1,2 - ,,(r)"1,,(.1') '" --------exp (J,r' + //2) (~)

\12'7r(;r:! + 1!:!)1/4

Unexpectcdly. the approximalion (8) appears even moreaccmale than (6).

The corrcsponding results are ploued in Fig. l. HercafterIhe routine be.,>sik [2] is lIscd rOl' calculation 01' the accllratemagnitudes 01' I,,(;t).

R('\'. Mex. Fú. 44 (3) (199X) 231-234

A NEW WII>E-RANGE APPROXIt>.lATlONOF MODIFIED BESSEL FUNCI'IONS IN TER.MS(lF ELEMENTARY FUNCTIONS 233

ud 1-w 0.20 ~

¿e .2.2

"'f;;;

ü 0.16 ee "~ 0.14 ou. 1;.Q 0.12 f o.w <two 0.10 . "Oal 1

~'O 0.083o W; 0.06

i5 5 •o 0.04 .~

'" 7 ;;;0.029 •• 9

.00 a:2 • 10 12 " " " 20 " • 10 12 " " " 20

X X

(~ (h)Fl(iURE 2. Approxima(ion 01' moditlcd Bcssel functions (a) and rclative cnors 01 approximation (o) for 1/ = 1,3,5,7,9. Explicit andapproximatc functions are showll by solid and dotled lines. respec(ivcJy. Approxima(ioll giwl1 hy Eq. (9) !las hren uscd. Thc digils inside (!leplot show lhe onlcrs of lhe functions.

•

X

.. "¿ 14.2~ 12e~ 10

1;.o.<t

"O-ewo.~ o;;;~ -2

.• ~t ~ _

0.0 O,S 1.0 1.S 2,0 2.S 3.0 3.S 4.0 4.S S,O•••3.5 4.0 4.5

•. '/-{x)

1.0 1.S 2.0 2,S 3.0

XD.'

(a) (1)

f-rc Ul{E 3. Approximatioll 01'modilicd Besscl funclions (a) anu rclativc crrors 01'apprcnimalioll (h) for 1/ = (J, 1/'2. Explicil and approximalcfUllctions are shown hy solid and dOllCUUnes. respectively. Approximation (9) with one fitting pararnetcr has heen uscd. The digits inside Iheplot shO\v (he orders nI"the functions.

The parameler n is chosen from considcration that lhe ap-proximale and cxplidt vallles 01"Ir/(.r) coincide al:1: = (J. The

In principie, Eq. (X) may he uscJ slarling fmm v 2: 1.As olle expecled Ihe error grows as v approaches lo unil, hutrelllains suilahle for sOl11eapplications. HO\vever lhis error

can he reducel! signifkanlly applying small Illoditkalions01' Eq. (X); namely. inlrodllcing a t1lling ramilletero. derenJ-ing on /1 only, as foilows:

1,,(.,,)

n

(Vl+,,2¡'r2-,,¡'r)" ( ., ")""-----------t'xp J.r- + v- ..,¡:s; (.,.2 + n)I/.'

[~r(,,+I)].I.~ II

U(Y)

results are sho\\'n in Fig. 2. Unfortunately. the approxima-lions (X) and (9) give relalively high errors for /,/ < 1 (max-imum error approaches 10 16{7r, as 1/ lcnds lO zero, as illus-trated in Fig. -' l. Ilowev~r, one can reduce these errors usinglhe folltm,'ing cOllsideralions. It lurns out that for O :s lJ < 1lhe formula (9) gives i1n approprialc approximation for allnon-negalive.r excluding some inlerval near lo l. whcre Iheerror has a sharp peak. Thus. olle Illay inlroduce sorne fittingfUllction that cOlTects the formula (9) for this narrow inter\'alol1ly. JI is possihlc [o cOllstruct such a fUllclion in many ways .and \ve use lhe j"ollmving olle:

(VI + ,,1 j.¡-' - "j.,.)"I,,(.r) "" /1 ., I .1 Pxp (/.,.1 + ",) ..,¡:s; (.¡-- + ni) /

¡i=[I+.,,"xp(-."-")j2.G]-I. "I=O(J'. (lO)

Re\'. Me.\". Fís. 44 (3) ( 199X) 211-234

D.KH. MOROZOV AND VV VOITSEKIIOVICHII

e"/>{x)

107

x.,

O

"#-¿.~••E.;¡olie.

'" o'O -1

~ .2&lJO .3

.~ -4~Qi -5a:

• 10, 7

x

234

'-''-o

•oE¡;o~..¡;••o

1Il

"¿:'O 02O

"

(~ 00FIGURE 4. Approximation of modificd Besscl functions (a) and rclativc errors 01'approximation for v = O, 1/2, 1,2,3. Explicit aod approx-¡mate functions are shown by solid aod dotted ¡¡nes. respeclively. Approximation (10) with two fitting parameters has hecll uscd. The digitsinside the plot show the orders of the functions.

Tilc calculatcd rcsulls. ohtaincd hy means 01' Eq. (10). areplolted in Fig. 4. Dne can see tha1 lhe maxirnum error appearswithin 5%.

011 a separate intervals on the J~semi-axis lhey can grow, re-maining inside 5%. Although the formula has been derivedror non-ncgativc arguments. an cxtcnsion over negative argu-mcnls is straightforward.

3. Conc!usions

It has hecn shown that an approximation formula suggcstcdrOl" modiflcd Bcsscl functions ¡vez;) can be used for aU 0011-

ncgativc f/ ami .r. Thc calculatcd rcsults have dcmonstratcdthal lhe dcviations from explicit values are small ami only

Acknowledgments

This work was supportcd by Consejo Nacional de Ciencia yTecnología (México) projccts 1020P-E9507 and 400340-5-2410PE.

1. \1. Aorarnowitz and J. Stegun. lfmuihook (~rMathematicalFWlclÍOIlS. (Dover Puhlications. Inc .. New York. 1965).

2. G.N. Watson. Theory 01 ResseL FunctiOfls . (Cambridge U.Prcss. Camhridge. 1948).

:J. \V.I1. Prcss. B.P. Flanncry. S.A. Teukolsky, and W.T. Vcttcrling.NUlllerical R('cipes in e . (Cambridge University Prcss, Cam-hridge. 19X8).

4. Y.L. Luke. Malhematical.fimctiolls ami rj¡eir al'lJr()Ximatio!/!;.(Acmlemic Press lile .. New York, 1(75).

5. G.A. Bakcr. Jr. 1:\.H'llfials (JI Pade al'l'roximationJ. (AcadcmicPrcss Ine .. Ncw York. 1(75).

G. I.S. Gradshtcin and Uv1. Ryzhik. Tablt, oIlmegrals, Series andProtillctS, (Academic Press. Ncw York. 1980). p. 958.

Rev. Mex. Fís. 44 (3) (1998) 231-234