Laplace Transform of Spherical Bessel Functions

Physica Scripta.Vol. 65, 369^372, 2002 Laplace Transform of Spherical Bessel Functions

A. Ludu1* and R. F. O’Connell2**

1Department of Chemistry and Physics, Northwestern State University, Natchitoches, LA 71497, USA 2 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA

Received May 14, 2001; revised version received October 6, 2001; accepted December 21, 2001 pacs ref: 02.30.Gp, 02.30.Uu

Abstract quantities are either the Fourier or Laplace transforms of We provide a simple analytic formula in terms of elementary functions for the mðtÞ. In particular, the blackbody radiation heat bath was ~ investigated in detail in Refs [2,3]. Laplace transform jl ðpÞ of the spherical Bessel function than that appearing in the literature, and we show that any such integral transform is a polynomial of Here, we wish to consider the one-dimensional Debye order l in the variable p with constant coe⁄cients for the ¢rst l À1powers,and model [4] because it leads to a result for the memory function with an inverse tangent function of argument 1=p as the coe⁄cient of the which involves spherical Bessel functions. power l. We apply this formula for the Laplace transform of the memory function related to the Langevin equation in a one-dimensional Debye model. In this model one takes all the heat bath oscillators to have thesamemassi.e.mj m for all j. Thus, taking the maxi- mum allowed heat bath frequency to be oL we see that 1. Introduction. The memory function in the Debye model Eq. (2) reduces to Z X oL In recent years, there has been widespread interest in dissi- 2 2 mðtÞ¼m oj cos ojt ¼ m doDðoÞo cos ot ð4Þ pative problems arising in a variety of areas in physics, j 0 and we refer to [1] for a mini-review. As it turns out, solutions of many of these problems are encompassed by a where [4] generalization of Langevin’s equation to encompass L DðoÞ¼ ; ð5Þ quantum, memory and non-Markovian e¡ects, as well as pv arbitrary temperature and the presence of an external potential VðxÞ. As in [2], we refer to this as the generalized quan- is the density-of-states. Also, v is the velocity of sound and L tum Langevin equation (GLE): is the length of a one-dimensional line such that o ¼ vk Z where Dk ¼ð2p=LÞ is the interval between the allowed values t 0 0 0 0 of the wave-vector k.Itfollowsthat mx€ þ dt mðt À t Þx_ðt ÞþV ðxÞ¼FðtÞþf ðtÞ; ð1Þ Z À1 mL oL mL ð Þ¼ 2 ¼ 3 f ð ÞÀ ð Þg where V 0ðxÞ¼dVðxÞ=dx is the negative of the time- indepen- m t doo cos ot oL j0 oLt 2j2 oLt : pv 0 3pv dent external force and mðtÞ is the so-called memory function. ð6Þ FðtÞ is the random (£uctuation or noise) force and f ðtÞ is a c-number external force (due to a gravitational wave, for Thus, in order to calculate m~ðpÞ we will turn to our general instance). In addition (keeping in mind that measurements resultgiveninthenextsectionandrelatedtoLaplacetrans- of Dx generally involve a variety of readout systems form of spherical Bessel functions. For example, for the involving electrical measurements), it should be strongly above case, the following expression for the Laplace trans- emphasized that ‘‘^ the description is more general than form is obtained the language ^’’ [2] in that xðtÞ can be a generalized displace- mL 2 À1 oL ment operator (so that, for instance, Dx could represent a m~ðpÞ¼ oLp À p tan ; ð7Þ voltage change). Furthermore, mðtÞ and FðtÞ are given in pv p terms of the parameters of the heat bath only. Explicitly from a more general formula. For a detailed discussion of X 2 this result we refer to Ref. [5]. mðtÞ¼ mjoj cosðojtÞyðtÞ; ð2Þ j where yðtÞ is the Heaviside step function. Also X 2. Analytic formula for the Laplace transform 2 h FðtÞ¼ mjoj qj ðtÞ; ð3Þ InordertocalculateEq.(7),aswellastheLaplacetransform j of higher order Bessel functions for other applications, we where qhðtÞ denotes the general solution of the homogeneous introduce in the following an exact formula, much simpler equation for the heat-bath oscillators (corresponding to no than formulas found in literature, in terms of trigonometric interaction). Thus, we have all the tools necessary for the functions and polynomials. analysis of any heat bath. As emphasized in Refs [1,3] of The spherical Bessel functions are given by (Ref. [6], primary interest for the calculation of observable physical p. 965)

1 *E-mail address: [email protected] p 2 jlðtÞ¼ Jlþ1ðtÞ; ð8Þ *E-mail address: [email protected] 2t 2

# Physica Scripta 2002 Physica Scripta 65 370 A. Ludu and R. F. O’Connell where l is a positive integer and Jn are the Bessel functions of the Fourier representation of the spherical Bessel functions the¢rstkindofrealargumentt. The Laplace transform, are de¢nite integrals over Legendre poynomials de¢ned by Z 1 Z 1 l ilx 1 jlðtÞ¼ ðÀiÞ e PlðxÞdx: ~ Àpt 2 À L½jlðtÞ jlðpÞ¼ jlðtÞe dt; ð9Þ 1 0 The following step is to perform the integration for canbecalculatedbyusingtherelation[Ref.[7],p.182,II(9)] ~ the Laplace transform, obtaining the relation jlðpÞ¼ ! lþ1 i QlðipÞ,whereQlðpÞ are the Legendre functions of second p Àn pffiffiffiffiffiffiffiffiffiffiffiffiffi type [6^8]. Pm p2 þ 1 However, none of these approaches can provide an ana- L½tmJ ðtÞ ¼ Gðm þ n þ 1Þ ; ð10Þ n mþ1 lytic expression in terms of elementary functions for the ðp2 þ 1Þ 2 l 2 case. Thus, we are motivated to develop a new n where Pm are the Legendre functions and we have to ful¢ll approach based on recursion relations. By di¡erentiating the restrictions: m þ n > À1andp >R 0. Also, G is the Gamma two times the second order spherical Bessel function 1 Àt xÀ1 function de¢ned [6] as GðxÞ¼ 0 e t dt for positive values of x. 3 sin t 2cos t j2 ¼ À 1 À ; ð16Þ In the case of the spherical Bessel functions we have t2 t t2 n ¼ l þ 1=2andm ¼À1=2 and hence the restriction is and by using recurrently the formula for the Laplace trans- l > À1 which is always ful¢lled since l ¼ 0; 1; .... Hence form of the derivative of a function f ðtÞ (Ref.[7],p.129)

rffiffiffi 1 p r 2 ÀlÀ1 ~0 ~ ~ ð Þ¼ ð þ Þ 2ð Þ f ðpÞ¼pf ðpÞÀf ð0Þ; ð17Þ jl p G l 1 PÀ1 r 2 p 2 where the prime denotes the derivative with respect to p,we lþ1 rffiffiffi 12 p Gðl þ 1Þ r 2 1 À r 2 ð11Þ can calculate the Laplace transform of j2:That is we choose ¼ 2 Gðl þ 3=2Þ p 1 þ r for f ðtÞ¼sin at=t which has its Laplace transform in tables [6,7]. We di¡erentiate f ðtÞ once and we calculate the Laplace 1 1 3 1 À r Â F ; ; l þ ; ; transform of f 0ðtÞ by using Eq. (17). Then, we di¡erentiate 2 2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi one more time and ¢nd again the Laplace transform of 00 where r ¼ p= p2 þ 1 and p > 0sothat1> r > 0. Here f ðxÞ by formula Eq. (17). Since we can express j2 only in FðÞa; b; g; x is the Gauss hypergeometric series de¢ned terms of the functions f ðtÞ and f 00ðxÞ we can express its by the formula [6] Laplace transform in terms of the Laplace transforms of f ðtÞ; f 00ðtÞ. We ¢nally obtain X1 ðkÞ aðkÞb zk FðÞ¼a; b; g; x Á ; 2 gðkÞ k! 3p 1 À1 1 3p k¼0 ~j ¼ þ tan À : ð18Þ 2 2 2 p 2 where aðkÞ ¼ aða þ 1Þ:::ða þ k À 1Þ; að0Þ ¼ 1. The formula provided by Eq. (11) is somehow di⁄cult to This procedure gives us a hint for constructing a recursion use in the cases l > 2, and requires tedious calculations formula for the Laplace transform of the spherical Bessel and further manipulations such as integration or other functions of any order. This will provide a much simpler transforms. However, in the case l ¼ 0, Eq. (11) reduces to form than that one provided by Eq. (11). This approach needs only the derivative recursion formula, Eq. (17), and ffiffiffi 1 À1 p r 2 1 þ r 4 1 1 3 1 À r the transform of the zero order function. By starting from ~j ðpÞ¼ 2 F ; ; ; : ð12Þ the recursion formulas for the spherical Bessel functions 0 p 1 À r 2 2 2 2 (Ref.[6],p.967) Since the Gauss hypergeometric series in Eq. (12) can be writen in the form (Ref. [6], p. 1041 II 13) 0 l jlþ1ðrÞ¼ÀjlðrÞþ jlðrÞ; ffiffiffi r À1p ð19Þ 1 1 3 sin r 2l þ 1 F ; ; ; r ¼ pffiffiffi ; ð13Þ j ðrÞ¼ j ðrÞÀj ðrÞ; l 1; 2 2 2 r lþ1 r l lÀ1 we ¢nd for the Laplace transform of j0ðtÞ¼sin t=t the usual and by writing them in a convenient form expression (Ref. [7], p. 152) 2l þ 1 0 l 1 jlþ1 ¼À jl þ jlÀ1; ð20Þ ~j ¼ tanÀ1 : ð14Þ l þ 1 l þ 1 0 p we ¢nd that any spherical Bessel function can be expressed as Another possibility for expressing the Legendre function for a sum of the derivatives of di¡erent orders of j0 the l ¼ 0 case in a simpler way is to use the result (Ref. [6], 2Xkl lÀ2k p. 1008) d j ðrÞ¼ CðlÞ j ðrÞ; ð21Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l k drlÀ2k 0 k¼0 À1 À1 À 2 P 2ðrÞ¼lim P 2 ðrÞ¼cos 1ðrÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi: ð15Þ ðlÞ À1 qÀ1 where C are coe⁄cients depending on the labels l and k.In 2 q!0 2 p 1 À r2 k order to calculate these coe⁄cients we use again the Finally, one can use a direct approach, based on the fact that recursion relation Eq. (20) and the formula in Eq. (21)

Physica Scripta 65 # Physica Scripta 2002 Laplace Transform of Spherical Bessel Functions 371 and we obtain the equation polynomial P of order l multiplied by the inverse tangent function of 1=p 2Xkl l 2kXlÀ1 CðlÞjðlÀ2kþ1Þ ¼ CðlÀ1ÞjðlÀ2kÀ1Þ k 0 2l þ 1 k 0 k¼0 k¼0 ~ À1 ð22Þ jl ¼ PlðpÞ tan ð1=pÞþQlÀ1ðpÞ: ð30Þ l þ 1 2kXlþ1 À Cðlþ1ÞjðlÀ2kþ1Þ; 2l þ 1 k 0 k¼0 Finally, we compare the above algorithm, based on where the superscript on the Bessel function indicates the recurrsion relations Eqs (24, 25) and Eqs (29, 30), with direct order of di¡erentiation. Since the Wronskian form of any numerical algorithms used in the corresponding built-in functions in di¡erent symbolic programs, like for instance two derivatives of j0 is not identical to zero, all these derivatives are functional independent so we can determine Mathematica. All functions involved above have their equiv- ðlÞ alent built-in correspondent in Mathematica-4.0 (used on a the numbers Ck in Eq. (22) by indentifying the coe⁄cients of the same order of derivative of the Bessel function. Conse- Power Macintosh 8500 computer), that is Gamma[z], quently we ¢nd the binary recursion relation BesselJ[n,z], LegendreP[n,z], LegendreQ[n,m,z], etc. While comparing the CPU time elapsed for both procedures 2l þ 1 we ¢nd out that our recurrsion approach is faster. That is the Clþ1 ¼À CðlÞ; ð23Þ 0 l þ 1 0 relative di¡erence between the two CPU intervals of time 3:2l which provide all k ¼ 0 coe⁄cients increases as e (numerical with respect to analytic), where l is the order of the spherical Bessel function. ð2l þ 1Þ!! Cðlþ1Þ ¼ðÀ1Þlþ1 ; l ¼ 0; 1; ... and Cð0Þ ¼ 0: 0 ðl þ 1Þ! 0 ð24Þ and a ternary recursion relation 3. Conclusions By using a generalization of the quantum Langevin’s ð þ Þ 2l þ 1 ð Þ l ð À Þ C l 1 ¼À C l þ C l 1 ; l ¼ 1; 2; ...; k l; equation in terms of memory function one can obtain exact k l þ 1 k l þ 1 kÀ1 solutions for dissipative problems arising in many areas ð25Þ in physics. In terms of a one-dimensional Debye model which provides the other coe⁄cients. Also, if we are given the memory function is expressed as a combination of ð2Þ C1 (whichcanbeeasilyobtainedbygivingparticularvalues spherical Bessel functions. However, of primary interest ð2Þ in Eq. (22), that is C1 ¼ 1=2) we can generate all the for the claculation of physical quantities are the Laplace coe⁄cients by using the ternary relation, Eq. (25). By using (or Fourier) transform of the memory function. this procedure it is easy to identify all coe⁄cients in the for- Also, the Schro« dinger equation for a free particle in polar mulaEq.(21).Forexamplewehave coordinates leads, for each value of the positive integer l of the orbital angular momentum, to a radial equation which 3 1 5 3 ð1Þ ð2Þ ð3Þ ð1Þ results in the generic equation for spherical Bessel functions. j1 ¼Àj0 ; j2 ¼ j0 þ j0; j3 ¼À j0 À j0 ; 2 2 2 2 ð26Þ In electrodynamics, too, the spherical Bessel functions are 35 ð4Þ 15 ð2Þ 3 related to solutions of the ¢eld equations in the stationary j4 ¼ j0 þ j0 þ j0; etc: 8 4 8 or quasi-stationary regime in cylindrical geometry. And The next step is to calculate the Laplace transform by using moreover, Bessel functions are involved in the Helmholtz Eqs (17) and (21). Since equation in cylindrical coordinates. In all such applications one needs the integral transforms (especially the Fourier ðÀ1Þk jð2kÞð0Þ¼ ; jð2kþ1Þð0Þ¼0; ð27Þ or Laplace transforms) of such solutions. Constructing an 0 2k þ 1 0 exact analytic formula for such applications is the object and from Eqs (14) and (17), we have of our study. Consequently, the present letter is not addressed to experts in special functions, but rather to 1 XnÀ1 physicists who just need to apply such formulas in their L½jðnÞ¼pn tanÀ1 À pnÀmÀ1jðmÞð0Þ; ð28Þ 0 p 0 research, in the simplest possible form. m¼0 We have obtained a simpler exact formula for the Laplace we ¢nally obtain the desired formula transform of the Spherical Bessel functions of any order, in " terms of polynomial and trigonometric functions, that is 1 2Xkl ~j ðpÞ¼pl tanÀ1 Á CðlÞpÀ2k in terms of elementary functions. Among other represent- l p k ations of such functions, this expression is much simpler, k¼0 3 lÀ1 ð29Þ hence more useful for potential applications. The speed 2Xkl ½ X2 Àk m ð Þ ðÀ1Þ of calculation is one of foremost importance for applications À C l pÀ2kÀ2mÀ15; k 2m þ 1 similar with the one presented above, especially in more than k¼0 m¼0 one dimension. We compared this formula with some valid for l 1, where ½Ástands for the integer part. The numerical evaluation in terms of precision and suitability Laplace transform is hence given in terms of elementary for faster implementation. Our comparison shows that functions, by a polynomial Q of order l À 1 in the variable the introduced formula is easy to be used and it introduces p, with constant coe⁄cients (all coe⁄cients determined less computational complexity than conventional numerical by the recursion relations Eqs (24) and (25)) plus another techniques.

# Physica Scripta 2002 Physica Scripta 65 372 A.LuduandR.F.O’Connell

Aknowledgements 3. Ford, G. W., Lewis, J. T. and O’Connell, R. F., Phys. Rev. Lett. 55, 2273 (1985). The authors are thankful to P. Abbott, The University of 4. Kittel, C., ‘‘Introduction to Solid State Physics’’, 7th edition (J. Wiley, Western Australia, for his valuable comments. One of us New York, 1996), pp. 118^120. (RFOC) would like to thank Professor Howard Lee for a 5. Kim, J. and Lee, M. H., Physica A, in press. The authors have obtained a preprint of Ref. 5, which motivated this paper. similar formula using the method of recurrence relations, which turns out to be much more complicated for the purpose of these calculations. 6. Ryshik, I. M. and Gradstein, I. S., ‘‘Tables of Series, Products and Inte- References grals’’, 4th edition (Academic press, New York, 1980). 7. Bateman, H., ‘‘Tables of Integral Transforms’’, Vol. I (McGraw-Hill 1. O’Connell, R. F., Intr. J. Quantum Chem. 58, 569 (1996). Book Company, Inc., New York, 1954). 2. Ford, G. W., Lewis, J. T. and O’Connell, R. F., Phys. Rev. A 37,44193 8. Abramowitz, M. and Stegun, I., ‘‘Handbook of Mathematical Func- (1998). tions’’, (Dover, 1970).

Physica Scripta 65 # Physica Scripta 2002