Frobenius Series Solution of Fuchs Second-Order. Ordinary Differential Equations. via Complex Integration

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1 Iteratioal Mathematical Forum, Vol. 9, 4, o., HIKARI Ltd, Frobeius Series Solutio o Fuchs Secod-Order Ordiary Dieretial Equatios via omplex Itegratio W. Robi Egieerig Mathematics Group Ediburgh Napier Uiversity olito Road, EH 5DT, UK opyright 4 W. Robi. This is a ope access article distributed uder the reative ommos Attributio Licese, which permits urestricted use, distributio, reproductio i ay medium, provided the origial work is properly cited. Abstract A method is preseted (with stard examples) based o a elemetary complex itegral expressio, or developig Frobeius series solutios or secod-order liear homogeeous ordiary Fuchs dieretial equatios. The method reduces the task o idig a series solutio to the solutio, istead, o a system o simple equatios i a sigle variable. The method is straightorward to apply as a algorithm, elimiates the maipulatio o power series, so characteristic o the usual approach [4]. The method is a geeraliatio o a procedure developed by Herrera [4] or idig Maclauri series solutios or oliear dieretial equatios. Mathematics Subject lassiicatio: 3B, 3E 34A5, 34A3 Keywords: Frobeius, Series solutio, Fuchs dieretial equatios, complex itegrals. Itroductio We cosider the Fuchs secod-order liear ordiary dieretial equatio (ODE) () () () ( ) + P( ) ( ) + Q( ) ( ) (.)

2 954 W. Robi with the superscript umbers i brackets deotig dieretiatio with respect to, the eroth derivative beig the uctio () itsel. As usual, we assume [4] p( ) ( ) P( ) q() ( ) Q( ) (.) are aalytic uctios o the idepedet variable, with beig a regular sigular poit o (.)..The class o liear ODE represeted by equatio (.) cotais may o the importat equatios o classical Mathematical Physics its solutio is o suiciet practical importace to warrat urther study. The usual [6, 4] Frobeius power series solutio or the liear ODE (.) is ( ) am( ) (.3) m m+ r with the ukow coeiciets { a m} m the idex r to be determied by substitutig (.3) ito (.), which gives rise to a recurrece relatio or the { a m } m. I particular, the idex r is obtaied as the solutio(s) o the idicial equatio r ( r ) + p() r + q() (.4) which arises orm the leadig term ( a ) o substitutig (.3) ito (.) [6, 4]. This, the stard approach, ivolves (i geeral) the maipulatio o iiite series. However, by a elemetary device, the ecessity or maipulatig iiite series directly, whe solvig liear ODE, ca be side-stepped, i the resultig ormat, maipulatios are reduced to solvig simple equatios alog with some basic algebra. The basis o this alterative approach to the determiatio o Frobeius power series, is the well-kow ormula rom complex variable theory [5] (all closedcotour itegrals that occur below are assumed evaluated i the couter-clockwise or positive directio) ( ) πi, d, otherwise (.5) where is a complex variable a ixed poit withi the closed-cotour. The relatio (.5) is used to kock-out all but oe term rom (.3) through a combied dieretiatio/itegratio process, described i detail i sectio, which leads to a complex itegral represetatio, equatio (.) below, or the coeiciets a o (.3). { m} m

3 Frobeius series solutio 955 Oce equatio (.) is established, it becomes a routie matter to apply it to the solutio o wide classes o ODE, especially those o the Fuchs class that are the mai topic o discussio here. Assumig a solutio o the Frobeius orm (.3), oe multiplies through the give dieretial equatio by ( ), applies a cotour itegratio (roud a particular closed-path) throughout the resultig expressio applies equatio (.), term by term. The result o this process, is that the origial dieretial equatio is trasormed ito the recurrece relatio or the coeiciets, { a }, o (.3), with the substitutio yieldig the idicial equatio tout court. (The process, or algorithm, trasorms the dummy variable m ito the dummy variable. ) From the operatioal poit o view, the basic problem o solvig the ODE is reduced, i practice, to the repeated solutio o a simple algebraic equatio i oe variable! From the historical poit o view, the method is a geeraliatio o a method origiated by Herrera [4] or dealig with Maclauri series solutios to oliear ODE. The path o the paper proceeds as ollows. I sectio we derive the basic ormula, (.), as a geeraliatio, to the case o a Frobeius power series, o Herrera s ormula or the coeiciets o a Maclauri series [4]. Next, i sectio 3, we apply the algorithm to some stard ODE rom mathematical physics: the Bessel equatio o order ν [, 4], the hypergeometric equatio [5, 4] Heu s equatio []. I these examples, i the others that ollow i sectios 4 5, we cosider the problem solved as soo as we have developed the recurrece relatio, the rest o the solutio process beig well-documeted i the literature. I sectio 4 the Riema-Papperit equatio is cosidered we see, explicitly, the eects o the stard trasormatio approach [3] to its solutio o the trasormatio o the recurrece relatios or the coeiciets. Fially, sectio 5 provides a brie discussio o the method (its geeric problems other applicatios) roud thigs o with a aother couple o examples. Note that, while we have applied the method to a stard rage o classical equatios; may other equatios will, aturally, yield a solutio to this approach. Further, we have restricted ourselves to the cosideratio o a idividual sigular poit i each example below: the sigular poit at the origi. Other sigular poits o the ODE may be trasormed to the origi, i ecessary.. Derivatio o the Basic Formula Progressig ormally, suppose we start with a assumed Frobeius power series expasio o a uctio, (), a regular sigular poit, that is [4] ( ) am( m ) m+ r (.)

4 956 W. Robi or costat idex r. The, with a closed-cotour aroud avoidig other sigularities o (), we wish to show that, or m k, k +, k +, k + 3, K, k,,,3,k a m [ m + r] k πi ( ) m+ + ( ) d (.) where, ollowig the otatio o Ice [6], or positive itegers k [ m + r] k Γ ( m + r + ) ( m + r)( m + r )( m + r ) L ( m + r k + ) (.3) Γ ( m + r k + ) while [ m + r ]. First, we dieretiate (.) k times, to id that or ( k m m k ) [ m + r] a ( ) m+ ( k k m m k ) [ + r] a ( ) + [ m + r] a ( ) m+ (.5) (.4) + Next, i we divide through (.5) by ( ) itegrate roud a closedcotour, cotaiig while avoidig sigularities o (), we get, as the idex r cacels-out ( ) + ( ) d [ + r] k a d [ + r] k ( ) a πi (.6) so that a [ + r] k πi ( ) + ( ) d (.7) or, chagig dummy variables, we have (.). Fially, lookig back, we id ( ) m k, k +, k +, k + 3, K, k,,,3, K. We ote that i r is ero, the (.) becomes the origial ormula o Herrera [4], that is a m ( m k)! m! πi ( ) m k + ( ) d (.8) We move o, ow, to some stard examples. We ote that series covergece is discussed, e.g., i [6], [7] [4] is ot cosidered here.

5 Frobeius series solutio The Solutio o Some Stard Secod-Order Equatios I this sectio, we will restrict the discussio to equatios with regular sigular poits at the origi. Notice that we will always write the geeral recurrece relatio i terms o a, which is as it must be writte [6, 4]. As a irst example, we solve the Bessel equatio o order ν, that is [, 6, 4] () () ( ) + ( ) + ( ν ) ( ) (3.) Next, Assumig (.), we divide through equatio (3.) by roud the closed-cotour, to get () () () () + () r+ itegrate ( ) ( ) ( ) ( ) + + d d d ν d (3.) r r r r compare the powers o the deomiators o the itegrs o (3.) with that o (.) to get our equatios or the dummy-variable m, oe or each value o k (two, oe, ero ero, respectively), that is m + r k + m + r + + r or m (3.3a) m + r k + m + r + + r or m (3.3b) m + r k + m + r + + r or m (3.3c) m + r k + m + r + + r + or m (3.3d) Havig idetiied the appropriate values o k m, we use (.), agai, to rewrite equatio (3.), ater cacellig re-arragig, as [( + r) ν ] a + a,,,,3,,k (3.4) Now, whe, as a, we must have a r ν (3.5) which is the idicial equatio, with solutios r ±ν. Also, whe, we must have a so we must set a. With a a arbitrary costat, we recogie (3.4) as the recurrece relatio(s) or J ν (). The third ial step is to solve the recurrece relatio (3.4) obtai the Frobeius series solutio explicitly. As this is well-kow [, 6, 4], we stop here Now, beore we cosider our ext example, we pause to poit-out that the idex r has, oce more, cacelled-

6 958 W. Robi out o the dummy-variable ( m ) calculatios. This appears to be a characteristic o the method as a whole (pace sectio ). For our secod example, we solve the hypergeometric equatio, that is [5, 4] ( ) () () ( ) + [ γ ( α + β + ) ] ( ) αβ ( ) (3.6) where α, β γ are costats. As beore, assumig (.), we divide through r+ equatio (3.6) by itegrate roud the closed-cotour, to get d + γ r r r+ () ( ) d () ( ) () () ( ) ( d αβ r r+ () ( ) d ) ( α + β + ) d (3.7) compare the powers o the deomiators o the itegrs o (3.7) with that o (.) to get ive equatios or the dummy variable m, oe or each value o k (two, two, oe, oe ero, respectively), that is m + r k + m + r + + r or m + (3.8a) m + r k + m + r + + r or m (3.8b) m + r k + m + r + + r + or m + (3.8c) m + r k + m + r + + r or m (3.8d) m + r k + m + r + + r + or m (3.8e) Havig idetiied the appropriate values o k m, we use (.), agai, to rewrite equatio (3.7), ater cacellig re-arragig, as ( + r + )( + r + γ ) a ( + r + α )( + r + β ) a (3.9a) or ( r)( + r γ ) a ( + r α )( + r β ) (3.9b) + a with,,,3, K. Now, whe, as a, we must have a, i this case, the idicial equatio is r ( r + γ ) (3.) so that r, or r γ. Apparetly, the irst root, r, will yield a ordiary power series. With a a arbitrary costat, we recogie, i (3.9), the recurrece ()

7 Frobeius series solutio 959 relatio(s) or the Frobeius series solutio o the hypergeometric equatio [5, 4], as beore, we termiate the example here. As our ial example i this sectio, we cosider the Frobeius series solutio o Heu s equatio, that is [] ( ) γ δ ε () αβ q () ( ) + [ + + ] ( ) + ( ) c ( )( c) (3.) with c, α, β, γ, δ, ε q costats where γ + δ + ε α + β +. learig ractios i (3.), we the multiply-out collect like terms to get 3 ( ( + c) + c) () ( ) + [( γ + δ + ε ) ( γ ( + c) + cδ + ε ) + cγ ] ( ) + ( αβ q) ( ) (3.) Next, assumig (.), we divide through equatio (3.) by roud the closed-cotour, assumig (.), to get ( + c) + d + c r r () ( ) d ( ) ( ) d r () ( ) ( ) + ( γ + δ + ε ) γ () () d ( γ ( + c) + cδ + ε ) + d + c r r r+ () + ( ) ( ) αβ d q d (3.3) r r+ () () () () r+ itegrate () ( ) d compare the powers o the deomiators o the itegrs o (3.3) with that o (.) to get eight equatios or the dummy variable m, oe or each value o k (two, two, two, oe, oe, oe, ero ero, respectively), that is m + r k + m + r + + r or m (3.4a) m + r k + m + r + + r or m (3.4b) m + r k + m + r + + r or m + (3.4c) m + r k + m + r + + r or m (3.4d) m + r k + m + r + + r or m (3.4e) m + r k + m + r + + r + or m + (3.4)

8 96 W. Robi m + r k + m + r + + r or m (3.4g) m + r k + m + r + + r + or m (3.4h) Havig idetiied the appropriate values o k m, we use (.), agai, to rewrite equatio (3.), ater cacellig re-arragig, as c( + r + )( + r + γ ) a + [( + r)[( + r + γ )( + c) + cδ + ε ] + q] + a or c ( + r + α )( + r + β ) (3.5a) a ( + r)( + r + γ ) a [( + r )[( + r + γ )( + c) + cδ + ε ] + q] a + a ( + r + α )( + r + β ) (3.5b) with,,,3, K, where we have used the act that γ + δ + ε α + β +. I this case, rom (3.5b), whe we must have a a ; also, as a c, the idicial equatio is (as i the previous example) r ( r + γ ) (3.6) so that r, or r γ. Agai, the irst root, r, will yield a ordiary power series. Equatio (3.5) costitutes the recurrece relatio(s) or the Frobeius series solutio o the Heu equatio (see [] reereces therei), as usual, we termiate the explicit calculatios here. However, we see that we have a threeterm recurrece relatio, (3.6), or the Heu Frobeius series solutio. For a brie discussio o this matter o three-term recurrece relatios see Figueiredo []. Further, i we set i (3.5b), we id a give i terms o a we have, ideed, oly oe solutio. 4. The Riema-Papperit Equatio I this sectio we cosider the Riema-Papperit equatio its relatio to the hypergeometric equatio [3]. To acilitate the discussio, we cosider, irst, the ollowig. The idicial equatio has its ow deiig dieretial equatio () () () ( ) ( ) + p ( ) ( ) + q ( ) (4.) To see this, we apply the method, with, to (4.), whe we id that (4.) trasorms to

9 Frobeius series solutio 96 r r ) a + p ra + q a (4.) ( or, as a, r r ) + p r + q (4.3) ( which is, ideed, the idicial equatio or (.). The equatio (4.) is just Euler s secod-order homogeeous equatio is obtaied (asymptotically) rom (.) whe (.) is rewritte as p P ) ( x) + pm( ( ) m Q q q ) ( x) + + qm( ( ) ( ) m m m (4.4a) (4.4b) the the leadig terms are extracted as. We are i a positio, ow, to examie the Riema-Papperit equatio, that is () A ( ) A A a b c () ( ) () B B B3 ( ) a b c ( a)( b)( c) (4.5) where a, b, c the A' s B' s are costats A + A + A3. There are three regular sigular poits at a, b c, but we will seek a Frobeius series solutio to (4.5) by irst movig the regular sigular poits to,. The argumet is a stard oe [3] ivolves the determiatio o the A' s B' s i terms o the roots o the idicial equatios correspodig to the three regular sigular poits at a, b c. So, i the roots o (4.3) correspodig to a are deoted by α, α, the roots correspodig to b by β,β the roots correspodig to c are by γ, γ, the, o shitig the sigular poits by takig a, b, c (4.6) the Riema-Papperit equatio may be rewritte, o clearig ractios, as [3] ( ) () ( ) + ( )( α α ( + β + β ) ) () ) + ( α α ( α α + β β γ γ ) + β β ) ( ) (4.7) () ( )

10 96 W. Robi I we apply the method to (4.7), or the regular sigularity at the origi ( a ), the we id, ater cacellig re-arragig, that (4.7) trasorms to ( + r α)( + r α) a [( + r α)( + r α) + ( a + ( + r + β )( + r + β) + γγ ] a + r + β )( + r + β ) (4.8) Settig i (4.8), we get the idicial equatio or (4.7) as ( r α )( r α ) a (4.9) as a a idetically. The relatio (4.9) provides a cosistecy check o the calculatio, as, with a, the roots o (4.9) must be α α, as required. As a urther cosistecy check o (4.8), without actually writig out the series, we ote, ollowig [3] agai, that (4.7) reduces to the hypergeometric equatio, (3.6), ater reductio, whe we choose α, α γ with β α, β β γ, γ γ α β (4.) I this reductio process, we expect (4.8) to reduce to (3.9) this is the case. Further, makig the substitutios (4.) i (4.8), we id that, ater some algebra ( + r)( + r + γ ) a ( + r + α)( + r + β ) a ( + r )( + r + γ ) a + ( + r + α)( + r + β ) a (4.) which is satisied i ( + r )( + r + γ ) a + ( + r + α)( + r + β ) a (4.a) as the ( r)( + r + γ ) a ( + r + α)( + r + β ) (4.b) + a also, we have retrieved the recurrece relatio (3.9b) or (3.6), as required. 5. oclusios Discussio While the moder tred is or solvig ODE usig computer algebra systems [3] (which is perectly atural ecessary i most cases) the acility with which the

11 Frobeius series solutio 963 preset method produces the coeiciets or Frobeius series solutios to liear ODE is exceptioable. The techical problems ivolved i idig the idicial equatio the recurrece equatio i the stard approach (see, e.g., [4]) are replaced, ow, with the solutio o simultaeous simple equatios, alog with the ecessary algebra, a act that holds true i geeral ot just or the examples preseted above. Ideed, i the examples preseted i the previous sectio we have cosidered oe sigular poit oly. But, as metioed above, whe other sigular poits exist, as with the hypergeometric Heu equatios, a chage o idepedet variable ( w ) trasorms these sigularities to the origi the method proceeds as beore (this applies to the poit at iiity also [9]). Further examples ca be geerated by cosiderig the coluet hypergeometric Heu equatios [5, 9] with the same type o procedure dealig with these equatios with equal acility as with the examples preseted i sectio 3. Additioally, as poited out by Hildebr [5], may secod-order equatios are special cases o M () M () M () ( + RM ) ( ) + ( P + PM ) ( ) + ( Q + Q ) ( ) M (5.) or particular choices o the costats M, P, PM, Q, QM RM. I act, special cases o (5.) iclude [5] the Bessel equatio, Legedre s equatio, Gauss s hypergeometric equatio, the coluet hypergeometric equatio, the Hermite equatio, the hebyshev equatio the Jacobi polyomial equatio. Naturally, it is possible to attack (5.) usig the preset approach also. At this poit i the discussio we tur to some other, related, matters i the solutio o (.). There are two immediate problems to discuss: the ecessity to solve, or at least deal with, three-or more-term recurrece relatios, the determiatio o the secod solutio to (.) which will ot, i geeral, be aother Frobeius series solutio, i the roots o the idicial equatio dier by a iteger [, 4]. I the case o three-term recurrece relatios, oe o the stard meas o avoidig three-term recurrece relatios is to trasorm ODE to a alterative orm, i the hope that the trasormed ODE has a power series solutio with a two-term recurrece relatio. For examples o this, see reerece [4]. Failig this, such three-or more-term recurrece relatios must be tackled o a case-by-case basis this is where computer algebra systems come back ito cosideratio. (See, also, reerece [] or a brie overview o the ideas behid the solutio o three-term recurrece relatios.)

12 964 W. Robi As or the determiatio o the secod solutio to (.), we have, at least i priciple, the geeral solutios to our equatios through the reductio o order method [4]. Alteratively, the special derivative method or idig a secod solutio (give a Frobeius series solutio), as explaied, or example, i the stard textbook o Raiville []. ca be attempted. Or, agai, it is possible to derive a ODE or the series part o the kow orm [4] o the secod solutio solve this usig the preset method. The actual choice o which method to attempt will deped o the particular circumstaces the details o the irst solutio. Naturally, the method preseted here ca be applied to higher-order ODE. Suppose we cosider the example o the l th -order liear homogeeous ODE with our sigularities aalysed, recetly, by Kruglov []. that is l i ( A i + B + ) i i i ( i) ( ) (5.) Applyig the method, we id that (5.) trasorms to l i ( i) ( i) ( i) ( ) ( ) ( ) A i d + B + + i d r i i d (5.3) r i r i+ so that, with l i k i usig (.), we get the geeral recurrece relatio ( A [ + r ] a + B [ + r ] a + [ + r] a ) (5.4) i i i i i i, whe, we get the idicial equatio (as a a while a ) l i [ ] (5.5) i r i we have Kruglov s scheme [], so we iish here. Fially, we wish to draw attetio to a certai amily relatioship betwee (.)/(.7) with r, the aputo ractioal derivative [8] D ( t) ( x) Γ ( k ) k + ( x t) x dt, >, x > ; k < < k, iteger k (5.6) though (5.6) is applied to dieret types o equatios i a dieret way [8].

13 Frobeius series solutio 965 Reereces [] Erdélyi A. (Editor): Higher Trascedetal Fuctios Volume II McGraw- Hill, New York (955). [] Figueiredo B. D. B.: Geeralied spheroidal wave equatio limitig cases. J. Math. Phys. 48 (7) 353. [3] Forsythe A. R.: Theory o Dieretial Equatios. (Volume 4) Part III Ordiary Liear Equatios. ambridge Uiversity press (9). [4] Herrera J..: Power series solutios i oliear mechaics. Brookhave Natioal Laboratory Report Udated. [5] Hildebr F. B.: Advaced alculus or Applicatios. Pretice-Hall, New Jersey (96). [6] Ice E. L.: Ordiary Dieretial Equatios. Dover, New York (956). [7] Jerey A.: Mathematics or Egieers Scietists, Fith Editio. hapma Hall, Lodo (996). [8] Kilbas A. A., Srivastava H. M. Trujillo J. J.: Theory Applicatio o Fractioal Dieretial Equatios. Elsevier, Amsterdam (6). [9] Kristesso G.: Secod Order Dieretial Equatios; Special Fuctios Their lassiicatio. Spriger, New York (). [] Kruglov V. E.: Solutio o the liear dieretial equatio o th -order with our sigular poits. Aales Uiv. Sci. Budapest., Sect. omp. 3 () [] Nikiorov A. F. Uvarov V. B.: Special Fuctios o Mathematical Physics. Birkhauser, Basel (988). [] Raiville E.D. Bediet P.E.: Elemetary Dieretial Equatios, Seveth Editio. Macmilla Publishig ompay, New York (989). [3] Seiler W. M.: omputer Algebra Dieretial Equatios - A Overview mathpad 7 (997) [4] Simmos G. F.: Dieretial Equatios with Applicatios Historical Notes, Secod Editio. McGraw-Hill, Lodo (99). Received: April 3, 4