New method for evaluating integrals involving orthogonal polynomials: Laguerre polynomial and Bessel function example

Transcription

1 New metho for evaluatig itegrals ivolvig orthogoal polyomials: Laguerre polyomial a Bessel fuctio eample A. D. Alhaiari Shura Coucil, Riyah, Saui Arabia AND Physics Departmet, Kig Fah Uiversity of Petroleum & Mierals, Dhahra 36, Saui Arabia haiari@mailaps.org Abstract: Usig the theory of orthogoal polyomials, their associate recursio relatios a ifferetial formulas we evelop a metho for evaluatig ew itegrals. The metho is illustrate by obtaiig the followig itegral result that ivolves the Bessel fuctio a associate Laguerre polyomial: e J( L ) ( ) Γ (si ) C (cos ), ( ) θ θ π where a are real parameters such that a >, is the Gegebauer (ultra-spherical) polyomial. 4 4 cosθ, a C ( ) Keywors: Defiite itegral, Bessel fuctio, Associate Laguerre polyomial, Gegebauer polyomial, recursio relatio. The use of spherical Bessel fuctios i the theoretical physics literature is overwhelmig. Oe reaso is that these fuctios are eige-solutios of the three imesioal free wave operator (the Laplacia) i spherical cooriates []. Aother is the fact that they make up the raial compoet of the wave fuctios for free particles i three imesios [,]. These free-particle wave fuctios are the referece i scatterig calculatios for iteractios with spherical symmetry. Cosequetly, the projectio of the Bessel fuctio oto various bases use i ifferet umerical schemes is importat. These weighte projectios (typically, itegrals) are ecessary for computig the epectatio values of selecte observables to be compare with measuremets. All itegral formulas havig the geeral form show i the Abstract, which oe coul fi i various tables of itegratio, are with the itegratio variable beig replace by i the argumet of the Bessel fuctio [3]. This is useful for may applicatios where the Gaussia bases (a.k.a. oscillator bases ) are wiely use i the calculatio [4]. The elemets of such a basis coul be writte as e L ( ), or a liear combiatio thereof. Noetheless, a equally useful basis, which is sometimes referre to as the Laguerre basis, has elemets of the form e L ( ). I typical scatterig problems with beig the raial cooriate a, where is the agular mometum quatum umber, the former basis is orthogoal whereas the latter is ot; it is trithogoal (i.e., the basis overlap matri the ietity represetatio is triiagoal). I this Letter, we use the theory of orthogoal polyomials, their associate recursio relatios a ifferetial formulas to evelop a metho for evaluatig a ew itegral that ivolves the Bessel fuctio J ( ) a the associate Laguerre polyomials L ( ). I all subsequet evelopmets we restrict our treatmet to real spaces a fiels. If J ( ) is the Bessel fuctio, the the spherical Bessel fuctio is efie as j ( ) π J ( ).

2 Sice J ( ) is efie o the positive real lie a the J ( ) lim ( ) Γ ( ) the we coul, i priciple, epa it as a ifiite series i terms of a complete set of basis fuctios which are compatible with the rage of J ( ) a its limitig values. Therefore, we assume that we ca make a separable epasio of J ( ) i the space spae by the square itegrable fuctios { ( ) } { c ( )}. That is, we set out to fi real fuctios { } c ( ) α J ( ) c ( ) ( ) c ( ) e L ( ) with real epasio coefficiets such that, () where, α >, a >. Usig the orthogoality property of the Laguerre polyomials [5] we ca write Γ ( ) ( α ) c( ) e J ( ) L ( ) ( ) Γ. () Therefore, we must impose the stroger costrait that > α >. The ifferetial equatio for the Bessel fuctio coul be writte as the eigevalue equatio D J( ) J( ), where D is the seco orer liear ifferetial operator [5]. Usig the ifferetial equatio of the Laguerre polyomials a their ifferetial formula, L L ( ) L, we ca write ( α ) ( ) D α α ( ). (3) We efie a cojugate space spae by the real L fuctios { ( ) } such that the basis overlap matri (i.e., the ier prouct m with the itegratio measure ) is triiagoal. That is, m if m. Thus, if we write ρ β ( ) e L( ), the the orthogoality of the Laguerre polyomials ictates that β α a ρ δ, (4) where δ or. Now, the matri represetatio of the eigevalue equatio D J ( ) ( ) coul be writte as J ( ) J c ( ) D D. (5) m m m Aitioally, we also require that the matri represetatio of the ifferetial operator D be at most triiagoal. That is, D m if m. Usig the actio of D o the basis give by Eq. (3) a the parameter relatio (4) this requiremet traslates ito a δ. Hece, we coul write I m ( D ) m ( α ) m α(m ) m (6) (α )( m ) m Agai, usig the orthogoality relatio of the Laguerre polyomials we obtai Γ ( ) Im ( )( α α) δ ( ) ( ) m α δ Γ, m (7) ( )( α α) δ m,

3 Therefore, Eq. (5) results i a three-term recursio relatio for the epasio coefficiets which is writte as I, c I, c I, c. I terms of the polyomials { P }, which are efie as Γ ( ) P( ) c ( ), (8) Γ ( ) this recursio relatio reas as follows ( )( α α) P ( )( α α) P (9) ( )( α ) P, The two-parameter polyomial, P α ( ), which satisfies this recursio relatio is, to the best of our kowlege, ew. However, for α Eq. (9) reuces to the three-term recursio relatio of the Gegebauer (ultra-spherical) polyomial C ( y ) [5], where 4 4 y y( ) cos θ a < θ π. Cosequetly, we ca write P( ) f ( ) C ( y), () where f ( ) is a arbitrary real fuctio of which is iepeet of the ie. Combiig Eq. () with Eq. (8) we get P( ) e J ( ) L ( ) ( ) ( ) f C y. () Therefore, what remais is oly to evaluate f ( ). To o that, we ifferetiate the above itegral with respect to usig the chai rule (i the sequece ) a the itegrate by parts sice the itegra vaishes at the e poits. The result is as follows: P ( ) ( ) e J L L L () e J ( ) L ( ) L ( ) L I goig from the first to the seco lie of this equatio, we have use the ifferetial formula a recursio relatio for L ( ). Thus, Eq. () states that P P ( ) P ( ) P. (3) O the other ha, ifferetiatig the right sie of Eq. () with respect to a usig ( y ) i the ifferetial relatio of the Gegebauer polyomial, y ( ) ( )( ) ( y ) C ( y) C ( y) C ( y), (4) y we obtai P f ( ) ( )( ) ( ) C P P. (5) ½ ½ Equatig the right ha sie of Eq. (3) with that of Eq. (5) a usig the recursio relatio of the Gegebauer polyomial we coclue that ( ) f ( ) ( ) y f ( ). (6) 3

4 Defiig g( y) f ( ) ( y ) y g yg givig the solutio ( ) f A (si θ ), where A is a costat which is iepeet of a siθ. 4, this coul be rewritte for g as ( ) Puttig all of the above together, we obtai the followig realizatio of the real epasio coefficiets c ( ) : Γ ( ) A c( ) (si θ) C (cos ) ( ) θ. (7) Γ To etermie A we substitute this i the epasio () a take the simultaeous limit ( ) ( ) ( ) C Γ, we obtai Γ ( ) Γ( ) ( ) A ( ) L () ( ) Γ Γ ( ) ( ) ( ) t L ep t t t A π f( ) A (si θ ) a. Usig ( ). (8), which is vali for Usig the geeratig fuctio ( ) t < [6], we get the followig value for the costat ( ) this i the epressio Γ. Isertig a substitutig i Eq. () we, fially, get the sought-after result e J( ) L ( ) ( ) (si ) (cos ) Γ θ C θ. (9) π I coclusio, we summarize the metho i the followig steps: ) We write J ( ) { ( )} as a ifiite series of proucts of fuctios, { } ( ) a c, that are compatible with the rage of efiitio of J ( ) a its limit values: Eq. () we ) Usig the orthogoality property of the polyomials i the basis fuctios { } write a itegral epressio for the epasio coefficiets { c } : Eq. () 3) We costruct the triiagoal cojugate space spae by { ( ) } a require that it also supports a triiagoal matri represetatio for the Bessel ifferetial operator D: Eq. (6) 4) The resultig three-term recursio relatio is solve i terms of orthogoal polyomials givig the epasio coefficiets { } c moulo a arbitrary real fuctio f ( ) : Eq. () 5) The fuctio f ( ) is etermie (up to a overall costat factor A ) by solvig a simple first orer liear ifferetial equatio which is obtaie by ifferetiatig the resultig itegral formula: Eqs. (-5) 6) The remaiig costat A is etermie by takig the limit of the epasio of J ( ) as (, ) : Eq. (8) 4

5 Oe shoul also ote that we eee kowlege of oly two properties of the Bessel fuctio to obtai the aalytic close-form result. These were the ifferetial equatio of J ( ) a its values at the limits. This is a highly otrivial observatio that coul have a major impact o the applicatio of the metho o a wier rage of fuctios. It as value to the metho a coul motivate its evelopmet ito a powerful itegratio tool for such problems. Ackowlegmets: The author is grateful to H. A. Mavromatis (Physics, KFUPM) a R. S. Alassar (Math, KFUPM) for fruitful cosultatios. Help i the calculatio provie by A. Al-Hasa (Physics, KFU) is highly appreciate. Refereces: [] P. M. Morse a H. Feshbach, Methos of Theoretical Physics, Vol. I (McGraw- Hill, New York, 953); R. Courat a D. Hilbert, Methos of Mathematical Physics, Vol. I (Itersciece, New York, 966); G. Arfke, Mathematical Methos for Physicists, e. (Acaemic, New York, 97) pp [] A. Messiah, Quatum Mechaics, Vol. I (North-Holla, Amsteram, 965); E. Merzbacher, Quatum Mechaics, e. (Wiley, New York, 97); R. L. Liboff, Quatum Mechaics, e. (Aiso-Wesley, Reaig, 99) [3] See, for eample, I. S. Grashtey a I. M. Ryzhik, Tables of Itegrals, Series, a Proucts (Acaemic, New York, 98) [4] R. G. Newto, Scatterig Theory of Waves a Particles, e. (Spriger, New York, 966); S. Gelma, Topics i Atomic Collisio Theory (Acaemic, New York, 969); B. H. Brase, Atomic Collisio Theory (Bejami, New York, 97); J. R. Taylor, Scatterig Theory (Wiley, New York, 97); P. G. Burke, Potetial Scatterig i Atomic Physics (Pleum, New York, 977) [5] A. Erélyi (E.), Higher Trasceetal Fuctios, Vol. I (McGraw-Hil, New York, 953); M. Abramowitz a I. A. Stegu (Es.), Habook of Mathematical Fuctios (Dover, New York, 965); T. S. Chihara, A Itrouctio to Orthogoal Polyomials (Goro a Breach, New York, 978); G. Szegö, Orthogoal polyomials, 4 th e. (Am. Math. Soc., Proviece, RI, 997) [6] W. Magus, F. Oberhettiger, a R. P. Soi, Formulas a Theorems for the Special Fuctios of Mathematical Physics, 3 r e. (Spriger, New York, 966) pp. 4 5