Tham khảo tài liệu 'mathematical method in science and engineering episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | GEGENBA UER FUNCTIONS TYPE A 153 GEGENBAUER FUNCTIONS TYPE A The Gegenbauer equation in general is given as 1 - - 2A l xdCị - n n 2A C x 0. dx dx For A 1 2 this equation reduces to the Legendre equation. For integer values of n its solutions reduce to the Gegenbauer or Legendre polynomials M 51 In the study of surface oscillations of a hypersphere one encounters the equation 1 _ x2 2 - 2m 3 xdU X XU x 0 LX InLt solutions of which could be expressed in terms of the Gegenbauer polynomials as u. x c x where A l m l m 2 . Using X COS0 u x ZJ ớ sinớ -m 1 we can put Equation into the second canonical form as Í5 í_ỈZÍS A m 1 2 lZr ớ 0- 9-257 av sin u On the introduction of A A m l 2 and comparing with Equation this is of type A with c p d 0 a 1 and z Ớ and its factorization is given by k 0 m m cot Ớ x m m2. 154 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD The solutions are found by using zr i d 7r-- 4JIi JLlI 2sin i0 1 Ự T ÓỊ and the formulas zp-1 z l 2-m2 1 2 Ị- -mcotớ zp Note that zp is the eigenfunction corresponding to the eigenvalue A z I 2 l-m 0 1 2 . that is to A I I 2 m I 2 z m l m 2 . SYMMETRIC TOP TYPE A The wave equation for a symmetric top is encountered in the study of simple molecules. If we separate the wave function as Ỉ7 ớ exp ÌKự exp jmp where ớ p and Ip are the Euler angles and n and m are integers G ớ satisfies the second-order ordinary differential equation dF t9 M - riZ e 8 e 0 9 266 where A w c and h are other constants that come from the physics of the problem. With the substitution Y Ớ sin1 2Ớ Equation becomes rf2y _ m - V2 m V2 2 - 2m cos 01 2 dớ2 sin2ớ K 1 4 1-11. BESSEL FUNCTIONS TYPE c 155 This equation is of type A and we identify the parameters in Equation as a 1 c 1 2 d K p 0. The factorization is now given by k G m m 1 2 cot G k sin G p m m 1 2 2. Eigenfunctions can be obtained from X sinJ c 1 2 Ẹ cosj