Tham khảo tài liệu 'mathematical method in science and engineering episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | COSMOLOGY AND GEGENBAUER POLYNOMIALS 73 sin2 Xd 2 sin y cosy w2 - H sin2yX y - -AX x . dx2 dx 0 jj and A are two separation constants. For wave problems w corresponds to the angular frequency. Two linearly independent solutions of Equation can be immediately written as T t eia t and e iajt while the Second Equation is nothing but the differential equation Ek . that the spherical harmonics satisfy with A and m given as A Z Z 1 z o 1 2 . and m o l . z. Before we try a series solution in Equation we make the substitution X x Co sin yC cosy where X cosy X 1 1 and obtain the following differential equation for C x 1-rr2 d C x dx2 - 21 3 x - ax 1 1 2 ứ2 - 2 0 C x 0. Substitution is needed to ensure a two-term recursion relation with the Frobenius method. This equation has two regular singular points at the end points X 1. We now try a series solution of the form C x akxk a k 0 to get Of Ct a l a _2 aia a l x -1 afc-i-2 fc a 2 k Ct 4-1 k 0 -ak k a k a-l 2l 3 k a - d 2 fc 0. In this equation A is defined as A -l l 2 2-Ậ fí2. n0 74 GEGENBAUER AND CHEBYSHEV POLYNOMIALS Equation cannot be satisfied for all X unless the coefficients of all the powers of X are zero that is Ooa a 1 0 do 0 ia a l o fe 2 ak k 4- a k a 1 2 3 fc a A k a 2 k a 1 k 0 1 2 . . The indicial Equation has two roots 0 and 1. Starting with the smaller root a 0 we obtain the general solution as Ti 2 2 21 3 A A A C x a0 1 - -ịx2 - ---- 34 ------- 2 r 2 4-3 3 1 x-ị------ where 0 and 1 are two integration constants and the recursion relation for the coefficients is given as fc fc -1 4- 21 3 k A . ak - ak ------- k 0 1 2 . fc 4-2 fc 4-1 J From the limit Jim 1 A co dfc we see that both of these series diverge at the end points X 1 as ---------77 1 To avoid this divergence we terminate the series by restricting Cư7 to integer values given by 11 N 0 1 2 . . Rũ Polynomial solutions obtained in this way
