ORTHOGONALITY OF LEGENDRE POLYNOMIALS

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1 olad robles of Applied Scieces, 05, Vol. 3, pp Szczeci dr Adrzej Atoi CZAJKOWSKI a, dr aweł IGNACZAK b a Higher School of Techology ad Ecooics i Szczeci, Iforatics ad Techical Educatio Wyższa Szkoła Techiczo-Ekooicza w Szczeciie, Edukacja Techiczo-Iforatycza b West oeraia Uiversity of Techology, Faculty of Ecooics, Departet of Applied Matheatics i Ecooics Zachodiopoorski Uiwersytet Techologiczy, Wydział Ekooiczy, Katedra Zastosowań Mateatyki w Ekooii ORTHOGONALITY OF LEGENDRE OLYNOMIALS Abstract Itroductio ad ai: The paper presets soe Legedre polyoials, orthogoality coditio for Legedre polyoials, recurrece forula ad differetial equatio for Legedre polyoials. The ai of the discussio was to give soe proof of orthogoality of Legedre polyoial syste. Material ad ethods: Selected aterial based o soe kowledge about Legedre polyoials which has bee obtaied fro the right literature. The proof of the theore describig the orthogoality of Legedre polyoials has bee elaborated usig a deductio ethod. Results: Has bee show soe proof of the theore describig the orthogoality of Legedre polyoials. It has bee show a exaple of orthogoality testig a pair of two arbitrary Legedre polyoials. Coclusios: I the paper has bee show the proof for theore: The syste of Legedre polyoials is orthogoal i the iterval, with the weightig fuctio p(z)=. Keywords: The syste of Legedre polyoials, theore of Legedre polyoials orthogoality, proof. (Received: ; Revised: ; Accepted: ) ORTOGONALNOŚĆ WIELOMIANÓW LEGENDRE A Streszczeie Wstęp i cel: W pracy przedstawioo wieloiay Legedre a, waruek ortogoalości dla układu tych wieloiaów, fukcję tworzącą oraz rówaie różiczkowe dla wieloiaów Legedre a. Cele rozważań było przeprowadzeie dowodu twierdzeia o ortogoalości układów wieloiaów Legere a. Materiał i etody: Materiał staowiły wybrae wiadoości o wieloiaach Legedre a uzyskae z literatury przediotu. W przeprowadzoy dowodzie zastosowao etodę dedukcji. Wyiki: okazao dowód twierdzeia o ortogoalości układów wieloiaów Legere a. odao przykład badaia ortogoalości pary dwóch dowolych wieloiaów Legedre a. Wiosek: W pracy przeprowadzoo dowód twierdzeia: Układ wieloiaów Legedre a jest ortogoaly w przedziale, z wagą p(z)=. Słowa kluczowe: Układ wieloiaów Legedre a, twierdzeie o ortogoalości układu wieloiaów Legedre a, dowód. (Otrzyao: ; Zrecezowao: ; Zaakceptowao: ) A.A. Czajkowski,. Igaczak 05 Coplex aalysis / Aaliza zespoloa

2 A.A. Czajkowski,. Igaczak. Aalytical fuctios ad orthogoal systes Defiitio.. The fuctio f (z) of the coplex variable specified i a certai regio D is called a aalytic fuctio i this area, where each poit of the doai D has a derivative f (z) []-[4]. Defiitio.. Two fuctios f (z) ad g(z) defied i the iterval a, b are called orthogoal fuctios i this iterval with the weight fuctio p(z) where the itegral of the product of three fuctios p(z), f(z) ad g(z) is equal to zero [5]: b p(z)f (z)g(z) dz = 0. () a Cosider the a syste of fuctios {f (z)} specified i the iterval a, b ad itegrable i it, the as we kow are also itegrable soe products of these fuctios take together with the weight fuctio p(z) as a third factor. Defiitio.3. If the fuctios of give syste are pairwise orthogoal with weight fuctio p(z), the the syste is called orthogoal syste of fuctios [7]. Defiitio.4. The syste fuctios {f (z)}, where = 0,,,... is called a orthogoal syste with weight fuctio p(z) i the rage a, b, if for each is true the followig equality [5]-[9]: b p(z)f (z)g (z) dz = 0 () a where p(z) is a predeteried o-egative fuctio that is idepedet fro the idicators ad kow as a weight fuctio.. Legedre polyoials Defiitio.. Legedre polyoials (z) for the value of a variable z are deteried by the forula [5]: d (z) = (z ) dla = 0,,, K. (3)! dz Let us copute a several Legedre polyoials ad write dow the geeral forula for the polyoial - (z). A few Legedre polyoials calculated directly fro the defiitio (3) is as follows: d = (4) 0! dz 0 0 0(z) (z ) = =,

3 Orthogoality of Legedre polyoials d (z) = (z ) = z = z, (5)! dz d d 4 (z) = (z ) = (z z +) =! dz 8 dz (6) d 3 = (4z 4z) = (z 4) = (3z ), 8 dz d 3 d 6 4 d 5 3 3(z) = (z ) = (z 3z + 3z ) = (6z z + 6z) ! dz 48 dz 48 dz d = (30z 36z + 6) = (0z 7z) = (5z 3z), 48 dz d 4 d 3 d 3 4(z) = (z ) = [4(z ) z] = [(z ) z] ! dz 384 dz 48 dz 3 d d 6 4 = (z 3z + 3z z) = (7z 5z + 9z ) = 3 48 dz 48 dz d 48 dz = (4z 60z + 8z) = (0z 80z + 8) = (35z 30z + 3), (z) = d (z ), (9) ( )! dz d (z) = (z ), (0)! dz ad so o. Theore.. (About the geeratig fuctio) (proof [], p. 63) Fuctio w(z, t) = () tz + t is the geeratig fuctio for Legedre polyoials, i.e. for sall values of t there is the followig series expasio [5]: w(z, t) = (z) t. () tz + t = 0 Theore.. (The secod order differetial equatio for Legedre polyoials) [4] (proof [], p. 67) If (z) are Legedre polyoials, the: [( z ) (z)] + ( + ) (z) = 0 dla = 0,,, K. (3) (7) (8) 87

4 3. Orthogoality of Legedre polyoials A.A. Czajkowski,. Igaczak Oe of the properties of Legedre polyoials is their orthogoality. Theore 3.. (Orthogoality of Legedre polyoials systes) [5] The syste of Legedre polyoials is orthogoal i the iterval, with weight fuctio p(z) =. roof: To deostrate the orthogoality of the Legedre polyoials syste should be show i accordace with the forula () that: p (z) dz = 0 (4) where ad p(z) =. We use here the differetial equatio for Legedre polyoials (3). We ultiply both sides of the differetial equatio (3) for -th Legedre polyoial by polyoial (z). The we get: + + = (5) [( z ) (z)] (z) ( ) (z) 0. Subsequetly we ultiply both sides of the differetial equatio (3) for -th Legedre polyoial by polyoial (z). The we get its followig for: + + = (6) [( z ) (z)] (z) ( ) (z) 0. Equatio (6) the we subtract by sides fro the equatio (5). Hece, we get: [( z ) (z)] (z) [( z ) (z)] (z) + + ( + ) (z) ( + ) (z) = 0. The above equatio (7) we reduce ito a for suitable for itegratio. Therefore, the equatio (7) after the idicated differetiatio with respect to variable z takes the followig for: [( z ) (z) + ( z ) (z)] (z) [( z ) (z) + ( z ) (z)] (z) + +[( + ) ( + )] (z) = 0. After the respective operatios equatio (8) takes the for: ( z ) (z) + ( z ) (z) ( z ) (z) + ( z ) (z) + + ( + ) (z) = 0. To the left side of equatio (9) we add ad subtract the expressio ( z ) (z), ad we use coutative law ad coectivity of added copoets law. Thus, the equatio (9) takes the followig for: [( z ) (z) + ( z ) (z)] + + [( z ) (z) + ( z ) (z)]+ +[ ( z ) (z) ( z ) (z)] ( ) (z) = 0. (7) (8) (9) (0) 88

5 Thus, After the appropriate operatios we get: Hece, Orthogoality of Legedre polyoials + ( z ) [ (z) (z)] ( z )[ (z) + (z)] ( z )[ (z) (z)] +[( + + ) ( + + )] (z) = 0. + ( z ) [ (z) (z)] + ( z )[ (z)] ( z )[ (z)] + + ( + + )( ) (z) = 0. {( z ) [ (z) (z)] + ( z )[ (z) (z)] }+ + ( + + )( ) (z) = 0. Fro which it follows that: {( z )[ (z) (z)]} + ( )( ) (z) = (4) The above equality (4) we itegrate both sides with respect to the variable z with i the rage of to +. Usig the appropriate properties of defiite itegrals we get: () () (3) {( z )[ (z) (z)]} dz + + ( + + )( ) (z)dz = 0dz. After applyig the appropriate properties of defiite itegrals i equality (5) we get: (5) + ( z )[ (z) (z)] ( )( ) (z)dz = (6) Fro the equality (6) shows that: ( )[ () () () ()] ( )[ ( ) ( ) ( ) ( )] + + ( + + )( ) (z)dz = 0. The first two copoets of equality (7) are equal to zero, therefore, we have: (7) ( + + )( ) (z)dz = 0 (8) where. Dividig the equality (8) o both sides by a factor is ot equal to zero (++)( ) 0 fially obtai: (z)dz = 0 (9) for, where weight fuctio p(z) =, which copletes the proof of the theore (3.). 89

6 A.A. Czajkowski,. Igaczak 4. Study of ay two orthogoal Legedre polyoials Let us ivestigate the orthogoal Legedre polyoials 3 (z) ad 4 (z) weight fuctio p(z)=, which are defied by the forulas (7) ad (8). Therefore: (z)dz = (5z 3z) (35z 30z + 3)dz. 8 (30) After polyoials ultiplicatio ad arragig we obtai: (z)dz = (75z 5z + 05z 9z)dz. 6 (3) After itegratio with respect to variable z we get: + (3) (z)dz = z z + z z After applyig the right properties of the defiite itegral we have: (z)dz = + +. (33) Fially, we get: 3 4(z)dz = 0. (34) Has thus bee show that Legedre polyoials 3 (z) ad 4 (z) are orthogoal i the iterval, with weight fuctio p(z) =. 5. Coclusio The syste of Legedre polyoials is orthogoal i the iterval, with the weight fuctio p(z) =. Literature [] Czajkowski A.A. (pod red.): robley Nauk Stosowaych robles of Applied Scieces, To / Volue, Wyd. Wyższej Szkoły Techiczo-Ekooiczej w Szczeciie, Szczeci 04. [] Fichteholtz G.M.: Rachuek różiczkowy i całkowy, To, WN Warszawa 976, w. 5. [3] Fichteholtz G.M.: Rachuek różiczkowy i całkowy, To 3, WN Warszawa 969, w. 3. [4] Krysicki W., Włodarski L.: Aaliza ateatycza w zadaiach, Część, WN Warszawa 970, w. 7. [5] Лебедев Н.Н.: Специальные функци и их приложения, Государственное Издательсто Физико-Математческой Литературы, Москва-Ленинград 963, издание второе. [6] Leja F.: Fukcje zespoloe, Biblioteka Mateatycza To 9, WN Warszawa 973, w. 3. [7] Sikorski R: Fukcje rzeczywiste, To, WN Warszawa 959, w.. [8] Sirow W.I.: Mateatyka wyższa, To 3, Część, WN Warszawa 965, w.. [9] Whittaker E.T, Watso G.N.: Kurs aalizy współczesej, Część, WN Warszawa 968, w.. 90

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