A Note on the form of Jacobi Polynomial used in Harish-Chandra s Paper Motion of an Electron in the Field of a Magnetic Pole.

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1 e -Joral of Sciece & Techology (e-jst) A Note o the form of Jacobi Polyomial se i Harish-Chara s Paper Motio of a Electro i the Fiel of a Magetic Pole Vio Kmar Yaav Jior Research Fellow (CSIR) Departmet of Mathematics V.S.S.D. (P.G.) College Kapr-8 U.P. (INDIA) viomaryaav@gmail.com Abstract: It is very iterestig a ifficlt to ersta the papers of the great mathematicia Prof. Harish-Chara (9-98). While reaig ay oe of them the reaer is compelle to ow the aswers of may qestios staig o the way. We have trie to ersta his paper a fo that the Jacobi polyomial appears o the way of soltio of wave eqatio of electro movig i the fiel of a magetic pole. This Jacobi polyomial is ot i the sal form appearig i mathematical literatre. I this paper we have compare the Jacobi polyomial se by Harish-Chara with its sal form. We have also ece the eplicit form for sch polyomials from ietity give i his paper. We have also verifie the reslts fo by him cocerig Jacobi polyomials. Keywors: Wave eqatio Jacobi polyomial Ietity Eigevales Eigefctios. Mathematics Sbect Classificatio Nmber: C45 47A75.. INTRODUCTION I his paper Prof. Harish-Chara obtaie the sitable Hamiltoia H for the motio of electro i the fiel of a magetic pole a rece the problem to fi the wave fctio satisfyig the wave eqatio (.) H E where E is some eigevale of H. The spherical polar cooriate system is sitable for the problem a therefore sig the trasformatio laws of tesor aalysis he coverte (.) to the followig form i r r r si (.) M cos im i cos. E Where he has writte e to mae the eqatio free from i.e. is a fctio of r oly a is mass of electro a M is half a o iteger. The two iepeet sets of Pali operators s a s satisfy the relatios (.) i i i i i i i i i i i a commte with all other operators ivolve i the eqatio (.). 55

2 e-joral of Sciece & Techology (e-jst) As i paper Harish-Chara assme that cos si (.4) K M cos becase the operator withi the crly bracet is prely imagiary. This operator commtes with the operator actig o i (.). This fact ca be verifie i the followig way. Beig the mltiple of the operator i the crly bracet i (.4) by i r the operator M cos cos commtes with i r si K. Frther K clearly commtes with E sice commtes with a. By brigig to the left of first term of the sqare we may write K as cot M cos cot M cos si si thi s maes easier to erstoo that it commtes with i r r. Ths we fi that K commtes with the operator of (.). Hece K eqal to the sqare of operator cos M cos si. Sice if two operators commte their eigevectors are same thogh their eigevales may be ifferet. Hece we ca choose to be a eigevector of K.. REDUCTION OF THE OPERATOR From eqatio (.4) K TO THE JACOBI OPERATOR cos K M cos si = cot M cos si cot M cos si = cot M cos si cot M cos si = cot M cos si () 7 56

3 e -Joral of Sciece & Techology (e-jst) si si cos M cos Sice cot si si si we get (.) cot cos si si cos M cos. K = M Now for fiig eigevectors of. K he pts cos si a First factor of R.H.S. of eqatio (.) cot cos si M = M = M = Also seco factor of R.H.S. of eqatio (.) M M. = si cos M cos = = Now eqatio (.) becomes M M K = M = 57

4 e-joral of Sciece & Techology (e-jst) = M M M M M M = M = M 4 4 (.) K = M 4 where cos. This is the reqire form of operator K.. EIGENVALUES AND EIGENFUNCTIONS OF OPERATOR K I paper Prof. Harish-Chara pt m M a say that if m are both itegral or both half-itegral the oly eigefctios of the operator (.) m correspoig to the iterval are the Jacobi polyomials P ( ) a correspoig eigevales are where is to be so chose that m a () 7 58

5 e -Joral of Sciece & Techology (e-jst) is a iteger. Prof. Harish-Chara efie Pm cos by ice a beatifl ietity which is oe of the achievemets of paper is give as (.) t cos t si t si t cos!! m m t t = P cos. m m! m! Ths from (.) we get (.) i.e. i.e. m P m ( ) m P m ( ) 4 P ( ) = = Pm ( ) 4 K P( ) = Pm ( ). So eigefctios of the operator K i the iterval polyomials P ( ) a the correspoig eigevales are m are the Jacobi. Also i.e. P = Pm ( ) K ( ) K Pm ( ) sice P ( ) is eigevector so K (.4) K. 59

6 e-joral of Sciece & Techology (e-jst) 4. EXPLICIT FORM OF P ( ) EVALUATED FROM HARISH-CHANDRA IDENTITY Harish-Chara Ietity (.) is a geeratig relatio for Jacobi polyomial P ( ). I paper Prof. Harish-Chara has ot give the eplicit form of P ( ). I this article we are givig sch form of P ( ) from Harish-Chara Ietity. Now from Ietity (.) which ca be writte as (4.) t si t cos t cos t si!! = m m m m!! m t t P cos t si t cos t cos t si L.H.S. of (4.) =!! =!! =!! r r tsi tcos r r t r s r s t rs rs s s tcos tsi s s r s rs s cos si pt r s m a cos cos cos si. L.H.S. of (4.) =!! mr m m t t r mr m r m m r r cos cos = m! m! m! m!!! m m t t r mr m r () 7 6

7 e -Joral of Sciece & Techology (e-jst) m m mr r r cos cos. Compare L.H.S. a R.H.S. of eqatio (4.) we get P m m! m!!! mr cos r mr r Hece we get (4.) P m which is a eplicit form of P ( ). m! m!!! r m m r cos cos. mr r mr r r m m r 5. JACOBI POLYNOMIAL IN ZEMANIAN From Zemaia chapter IX we ow that the ormalize eigefctios of the operator (5.) η = w D w D w where w = < < are give as (5.) a are real mbers with > > a w = h P... where h! a the P are the Jacobi polyomials efie as (5.) m m m m m P these eigefctios Ths we have correspo to the eigevales η 6 =.

8 e-joral of Sciece & Techology (e-jst) w D w D w i.e. = i.e. D D i.e. D = i.e. (5.4) So 4. are the eigevectors a ifferetial operator are eigevales of the. 6. COMPARISON OF TWO JACOBI POLYNOMIALS USED IN PAPER AND ZEMANIAN Accorig to Zemaia from eqatio (5.4) pt (6.) m we get m = () 7 6

9 e -Joral of Sciece & Techology (e-jst) m (6.) = m =. Comparig eqatio (.) a (6.) with sig (4.) (5.) a (6.) we get (6.) m Pm ( ) cos m (6.4) m Pm ( ) cos. m m Where be the ormalize Jacobi polyomial. 7. CONCLUSION We have come to the coclsio that the Jacobi polyomial P ( ) is ifact the m vale of for m a m. This allows s to tae a either both itegers or both half of o itegers. ACKNOWLEDGEMENT I am gratefl to my spervisor Dr. T. N. Trivei for giace a ecoragemet rig the sty of this wor. I am also thafl to CSIR New Delhi Iia for proviig fiacial spport. REFERENCES. Harish-Chara (948). Motio of a Electro i the Fiel of a Magetic Pole. Physical Review 74(8) : Dirac P. A. M. (958). Relativistic theory of the electro I : The Priciples of Qatm Mechaics Clareo Press Ofor Forth eitio p Zemaia A. H. (968). Trasformatios arisig from orthoormal series epasios I : Geeralize Itegral Trasformatios Itersciece Pblishers a ivisio of Joh Wiley & Sos Ic. vol. XVIII p

Definition 2.1 (The Derivative) (page 54) is a function. The derivative of a function f with respect to x, represented by. f ', is defined by

Definition 2.1 (The Derivative) (page 54) is a function. The derivative of a function f with respect to x, represented by. f ', is defined by Chapter DACS Lok 004/05 CHAPTER DIFFERENTIATION. THE GEOMETRICAL MEANING OF DIFFERENTIATION (page 54) Defiitio. (The Derivative) (page 54) Let f () is a fctio. The erivative of a fctio f with respect to,