Special Functions. Leon M. Hall. Professor of Mathematics University of Missouri-Rolla. Copyright c 1995 by Leon M. Hall. All rights reserved.

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1 Specil Fucios Leo M. Hll Professor of Mhemics Uiversiy of Missouri-Roll Copyrigh c 995 y Leo M. Hll. All righs reserved.

2 Chper 5. Orhogol Fucios 5.. Geerig Fucios Cosider fucio f of wo vriles, ( x,), d is forml power series expsio i he vrile : fx, ( ) = g( x. ) = The coefficies i his series re, i geerl, fucios of x, d we c hi of hem s hvig ee f geered y he fucio f. I fc, g( x) = ( x, ), hough here my e eer wys o compue! hem. If his ide is exeded slighly, we ge he followig defiiio: Defiiio 5... The fucio F( x,) is geerig fucio for he sequece { g( x) } if here exiss sequece of coss { c} such h F( x,) = c g ( x). = I is o ucommo for ll he cs o e oe. Oe of he pricipl prolems ivolvig geerig fucios is deermiig geerig fucio for give se or sequece of polyomils. Especilly desirle is geerl heory which c e used o ge geerig fucios. Uforuely, o such heory hs ye ee developed, so we mus e coe wih resuls for specil cses foud usig mipulive dexeriy. Exmple 5... Oe specil cse is whe he coefficies re successive powers of he sme fucio. Le { g( x) } = {( f( x)) }. The he geerig fucio c e foud usig he formul for he sum of geomeric series. F( x,)= ( f( x)) = f( x) provided fx ( ) <. = Exercise 5... Fid he geerig fucio for he sequece { ( f( x)) }. My ses of elemery d specil fucios hve ow geerig fucios. Here re some exmples. Exmple 5... The Beroulli fucios. Le yx, ( ) = B( x). Termwise differeiio wih respec o x d properies of he Beroulli fucios (secio.3) yields y ( x,) = y( x,). Thus, = [ ] x x e y( x, ) = e [ yx( x, ) y( x, )] =, x x d so for ech here is fucio C() such h y( x,)= C() e, d we hve x Ce () = B( x. ) (5).. Iegrio of (5..) over he iervl [, ] d properies of he Beroulli fucios give = C ()= e 55. x

3 e Thus, he geerig fucio for he Beroulli fucios is yx, ( ) =. e x Exercise 5... Fill i he deils i Exmple 5... Exmple Legedre polyomils: ( x + ) = P ( x). = Exercise Use Tylor s heorem o verify he firs hree coefficies i he geerig fucio relio for he Legedre polyomils. Exmple [ ] exp z J z ( Bessel fucios: ) = ( ). = Exmple Hermie polyomils. Deoe he Hermie polyomil of degree y H( x). The ( ) H( x) exp x =.! = Exercise Fid he firs four Hermie polyomils. Exercise Prove he expsios e e ( ) H ( x) cos x = ( )! = ( ) H ( x) si x = ( + )! = + + for <. These expressios c e hough of s geerig fucios for he eve d odd Hermie polyomils. The geerig fucios for oh he Legedre d he Hermie polyomils re fucios of he form G( x ). The followig heorem is represeive of heorems which give properies commo o ll ses of fucios hvig geerig fucios of his form. Theorem 5... If G( x ) = g ( x), he g ( x) = d, for, he g s sisfy he differeildifferece equio = xg ( x) g ( x) = g ( x ). Proof. Le F( x,) = G( x ) = g( x). The F sisfies he PDE ( x ) F F =. I x = erms of he series, his PDE is xg ( x) g ( x) = g ( x). = = = Equig coefficies gives he desired resul. 56

4 Exercise I A&S, pges 783-4, umer of geerig fucios re give s fucios of x +. Formule d prove he equivle of Theorem 5.. usig R i plce of x. R = 5.. Orhogoliy Cosider he DE ( x) y + ( x) y +[ ( x)+ λ] y =. ( x) ( x) ( x) px ( ) ( x) ( x) Muliply y he iegrig fcor px ( ) = exp dx, le qx ( ) = px ( ), d rx ( ) =, o ge he DE io he form [ px ( ) y] + [ qx ( ) + λrx ( )] y =. (5.. ) Equio (5..) is sid o e i Surm-Liouville form d if pproprie oudry codiios re specified o iervl we hve Surm-Liouville prolem. Vlues of λfor which Surm-Liouville prolem (SLP) hs orivil soluios re clled eigevlues of he SLP d he correspodig soluios re clled eigefucios. These ides re sudied i deil i courses o pril differeil equios d oudry vlue prolems where he SLP rises urlly i he soluio of PDEs wih oudry codiios. The followig heorem, sed here rher vguely, is proved i such courses. Theorem 5... Uder pproprie codiios, if ym d y re eigefucios correspodig o disic eigevlues of he SLP ssocied wih (5..) o he iervl [, ], he r( x) y ( x) y ( x) dx =. (5.. ) m Whe equio (5..) holds, we sy h ym d y re orhogol wih respec o he weigh fucio r( x). This equio c, i fc, e e s he defiiio of orhogoliy. Exercise 5... Wh do you cll ordo he Keucy Dery? Exmple 5.. (Legedre). Legedre s DE, s we hve see, is Surm-Liouville form, his ecomes [ ] ( x ) y + ( + ) y =. ( ) + ( + ) = x y xy y Here, px ( ) = x, qx ( ),, rx ( ), d λ= ( +). Sice x= ± re regulr sigulr pois, we c e sure soluios exis o he closed iervl [, ] oly whe he soluios re polyomils, so he eigevlues re =,,,... d he eigefucios re he correspodig Legedre polyomils P( x ),P( x ),P( x ),... By Theorem 5.. we hve P ( x) P ( x) dx = wheever m d re disic oegive iegers. m. I The orhogoliy iegrl is geerlizio o fucios of he do produc for vecors, d sice he do produc of vecor wih iself is he squre of he legh of he vecor, he iegrl i (5..) wih oh eigefucios he sme c e ierpreed s he legh squred of he eigefucio. Ofe, we w his legh o e oe for ll he eigefucios, i which cse we sy h he eigefucios re ormlized. Sice he eigefucios re orhogol y (5..), if hey re lso ormlized, we sy hey re orhoorml. 57

5 Exmple 5.. (coiued). We ow deermie P ( x) dx usig he geerig fucio. ( x + ) = P ( x) [ ] ( x + ) = P ( x) = + log = P( x) dx = ( ) = P( x) dx Equig coefficies gives he ormlizig coss for he Legedre polyomils: P = = ( x) dx =. + Exercise 5... Fill i he deils i Exmple 5... Exmple 5.. (Bessel). Bessel s DE of order, (slighly modified - do you see how?) x y + xy + ( λx ) y=, wrie i Surm-Liouville form is [ ] [ xy ] + λ x y =. x α For he iervl [,], he eigevlues re λ =, where α is he posiive zero of J( x). The orhogoliy iegrl is, for m =, xj ( λ x) J ( λ x) dx =. m Noe here h is fixed, d he differe eigevlues d eigefucios re deoed y he suscrips o λ or α. h For ses of polyomils, he followig equivle codiio for orhogoliy is ofe useful. We cll se of polyomils simple if he se cois excly oe polyomil of ech degree; uless sed oherwise, he degree of suscriped polyomil is equl o is suscrip. Theorem 5... If { φ( x) } is simple se of rel polyomils d r( x ) > o iervl (,), he { φ ( x) } is orhogol se wih respec o he weigh fucio r( x) if d oly if for =,,,...,, rx ( ) x φ( x) dx =. Oulie of Proof. The proof is sed firs o he fc h y polyomil of degree m<c e wrie m s lier comiio of powers of x from x hrough x. The he fc h x c e expressed s lier comiio of φ ( x) hrough φ ( x) is used. Deils re lef o he sude. 58

6 Exercise Prove Theorem 5... Exercise Prove h if { φ( x) } is simple se of rel polyomils d r( x ) > o iervl (,), he for every polyomil Pof degree less h, r( x) φ( x) P( x) dx =. Also prove h rx ( ) x φ( x) dx=. The ieresig pr of polyomil is er he zeros. Afer he ls zero d efore he firs oe, polyomils re rher orig - hey eiher go up, up, up, or dow, dow, dow. Theorem If { φ( x) } is simple se of rel polyomils, orhogol wih respec o weigh fucio r( x ) > o iervl (, ), he, for ech, he zeros of φ re disic d ll lie i he iervl (, ). Proof. For >, y Theorem 5.. r( x) φ ( x) dx=, so he iegrd mus chge sig les oce i (, ), d sice r( x ) >, his mes φ( x) chges sig i (, ). Le { α} s = e he se of pois where φ( x) chges sig i (,). These re he zeros of φ of odd mulipliciy, d sice he degree of φ is, we ow h s. Form he polyomil s P( x) = ( x α ). = Assume s<. The y Exercise 5..4, r( x) φ ( x) P ( x) dx =. Bu ll he zeros of φ( x) P( x) re of eve mulipliciy, so r( x) φ( x) P( x) co chge sig i (,). Hece, s<is o possile, d we mus hve s=. This mes h φ hs roos of odd mulipliciy i (, ). Sice he degree of φ is, ech roo is simple, d he heorem is proved Series Expsios A impor pplicio of orhogol polyomils i physics d egieerig is he expsio of give fucio i series of he polyomils. For simple se of polyomils, he powers of x i he usul series represeio re replced y he polyomils of pproprie degree. Of course, he prolem is o fid he coefficies i such series expsio, d his is where orhogoliy ecomes quie useful. Exmple Le f e defied i he iervl (, ), d expd f( x) i series of Legedre polyomils. I oher words, we w o deermie he coefficies i fx ( ) = c P( x ) (5.. 3 ) = so h equliy holds for x (, ). Proceedig formlly, we muliply oh sides y Pm( x) d iegre from o. c, f( x) Pm( x) dx = c Pm( x) P( x) dx = m = m+ 59

7 which implies h, for =,,,... c =( + ) f( x) P( x) dx. (5. 3. ) This procedure is e, cle, d lgorihmic, u we oo some mhemicl lieries which should les e cowledged. I priculr, how did we ow h fx ( ) could e represeed s i (5.3.) i he firs plce, d lso, ws i legiime o ierchge he operios of iegrio d summio? Uless hese pois re clered up, we hve o guree, excep fih, h (5.3.) wih coefficies give y (5.3.) coverges d hs sum fx ( ). Aoher cocer is h eve if we c e sure he procedure wors for Legedre polyomils, will similr procedure e vlid for differe se of simple orhogol polyomils? Foruely, for give se of orhogol polyomils, here re codiios which do guree h equios (5.3.) d (5.3.) or heir equivles re vlid. Uforuely, he codiios re differe for differe ses of polyomils. Proofs ge somewh ivolved, d re omied here, u ieresed reders my cosul Leedev or Whier d Wso. Theorem If he rel fucio f is piecewise smooh i he iervl (, ) d if f ( x) dxis fiie, he he Legedre series (5.3.) wih coefficies give y (5.3.) coverges o fx ( ) wherever fis coiuous. If x is poi of discoiuiy, he series coverges o he verge of he righ-hd d lef-hd limis of fx ( ) x. Exercise Expd fx ( ) = x i series of Legedre polyomils. {, x < α Exercise Expd fx ( ) = i series of Legedre polyomils, d verify he, α < x vlue x = α. Exercise Express fx ( ) = usig he geerig fucio. x i series of Legedre polyomils. Clcule he coefficies y I is possile o derive ll properies of se of orhogol polyomils y srig wih oly he geerig fucio. The followig series of exercises uilds up some resuls ou he Hermie polyomils defied i Exmple Exercise Show h he geerig fucio F( x,) for he Hermie polyomils sisfies F = x F, d so H ( x) = H ( x). Similrly, show h F( x,) sisfies ( x ) F =, d so H ( x) xh ( x)+ H ( x ) =. ( ) + F Exercise Show h he Hermie polyomils sisfy he differeil equio (Hermie s DE) y ( x) xy ( x)+ y( x ) =. Wrie Hermie s DE i Surm-Liouville form d deermie he iervl d he weigh fucio for he orhogoliy of he Hermie polyomils. x Exercise I his exercise, you will clcule e H ( x) dx. Begi y replcig he idex i (5.3.3) y d muliply y H ( x). The from his equio surc (5.3.3) muliplied y H ( x). 6

8 Wor wih his resul o oi e H ( x) dx = e H ( x) dx x x for =, 3,... Repeed pplicio of his reducio formul gives, for =, 3,..., Filly, show y direc clculio h (5.3.4) lso holds for =,. x e H ( x) dx =! π. ( ) There is resul for Hermie polyomils correspodig o Theorem 5.3., i which he iegrl required o x e fiie is e f ( x) dx. Exercise {, x < Expd fx ( ) = sgx ( ) =, x > i series of Hermie polyomils. 6

9 Refereces [] Armowiz, M., d Segu, I. (eds.), Hdoo of Mhemicl Fucios, Dover, New Yor, 965 [] Agew, R., Alyic Geomery d Clculus, wih Vecors, McGrw-Hill, New Yor, 96 [3] Jhe, E., d Emde, F., Tles of Fucios, Dover, New Yor, 945 [4] Leedev, N., Specil Fucios d Their Applicios, Dover, New Yor, 97 [5] Lue, Y., The Specil Fucios d Their Approximios ( vols.), Acdemic Press, New Yor, 969 [6] Mile-Thompso, L., Jcoi Ellipic Fucio Tles, Dover, New Yor 95 [7] Riville, E., Specil Fucios, McMill, New Yor, 96 [8] Riville, E., Iermedie Course i Differeil Equios, Wiley, New Yor, 943 [9] Spiegel, M., Theory d Prolems of Advced Clculus, Schum, New Yor, 963 [] Whier, E. d Wso, G., A Course of Moder Alysis, Cmridge, Lodo, 963 6