Mathematical Methods for Physics and Engineering - Matematica.NET

18.7 LAGUERRE FUNCTIONSit has a regular singularity at x = 0 and an essential singularity at x = ∞. Theparameter ν is a given real number, although it nearly always takes an integervalue in physical applications. The Laguerre equation appears in the descriptionof the wavefunction of the hydrogen atom. Any solution of (18.107) is called aLaguerre function.Since the point x = 0 is a regular singularity, we may find at least one solutionin the form of a Frobenius series (see section 16.3):y(x) =∞∑a m x m+σ . (18.108)m=0Substituting this series into (18.107) and dividing through by x σ−1 ,weobtain∞∑[(m + σ)(m + σ − 1) + (1 − x)(m + σ)+νx] a m x m =0.m=0(18.109)Setting x = 0, so that only the m = 0 term remains, we obtain the indicialequation σ 2 = 0, which trivially has σ = 0 as its repeated root. Thus, Laguerre’sequation has only one solution of the form (18.108), and it, in fact, reduces toa simple power series. Substituting σ = 0 into (18.109) and demanding that thecoefficient of x m+1 vanishes, we obtain the recurrence relationa m+1 =m − ν(m +1) 2 a m.As mentioned above, in nearly all physical applications, the parameter ν takesinteger values. Therefore, if ν = n, wheren is a non-negative integer, we see thata n+1 = a n+2 = ···= 0, and so our solution to Laguerre’s equation is a polynomialof order n. It is conventional to choose a 0 = 1, so that the solution is given byL n (x) =[x (−1)n n − n2n! 1! xn−1 + n2 (n − 1) 2]x n−2 − ···+(−1) n n! (18.110)2!n∑= (−1) m n!(m!) 2 (n − m)! xm , (18.111)m=0where L n (x) is called the nth Laguerre polynomial. We note in particular thatL n (0) = 1. The first few Laguerre polynomials are given byL 0 (x) =1, 3!L 3 (x) =−x 3 +9x 2 − 18x +6,L 1 (x) =−x +1, 4!L 4 (x) =x 4 − 16x 3 +72x 2 − 96x +24,2!L 2 (x) =x 2 − 4x +2, 5!L 5 (x) =−x 5 +25x 4 − 200x 3 + 600x 2 − 600x + 120.The functions L 0 (x), L 1 (x), L 2 (x) andL 3 (x) are plotted in figure 18.7.617