Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.7 LAGUERRE FUNCTIONS

it has a regular singularity atx= 0 and an essential singularity atx=∞.The

parameterνis a given real number, although it nearly always takes an integer

value in physical applications. The Laguerre equation appears in the description

of the wavefunction of the hydrogen atom. Any solution of (18.107) is called a

Laguerre function.

Since the pointx= 0 is a regular singularity, we may find at least one solution

in the form of a Frobenius series (see section 16.3):

y(x)=

∑∞

m=0

amxm+σ. (18.108)

Substituting this series into (18.107) and dividing through byxσ−^1 ,weobtain

∑∞

m=0

[(m+σ)(m+σ−1) + (1−x)(m+σ)+νx]amxm=0.

(18.109)

Settingx= 0, so that only them= 0 term remains, we obtain the indicial

equationσ^2 = 0, which trivially hasσ= 0 as its repeated root. Thus, Laguerre’s

equation has only one solution of the form (18.108), and it, in fact, reduces to

a simple power series. Substitutingσ= 0 into (18.109) and demanding that the

coefficient ofxm+1vanishes, we obtain the recurrence relation

am+1=

m−ν

(m+1)^2

am.

As mentioned above, in nearly all physical applications, the parameterνtakes

integer values. Therefore, ifν=n,wherenis a non-negative integer, we see that

an+1=an+2=···= 0, and so our solution to Laguerre’s equation is a polynomial

of ordern. It is conventional to choosea 0 = 1, so that the solution is given by

Ln(x)=

(−1)n

n!

[

xn−

n^2

1!

xn−^1 +

n^2 (n−1)^2

2!

xn−^2 −···+(−1)nn!

]

(18.110)

=

∑n

m=0

(−1)m

n!

(m!)^2 (n−m)!

xm, (18.111)

whereLn(x) is called thenthLaguerre polynomial. We note in particular that

Ln(0) = 1. The first few Laguerre polynomials are given by

L 0 (x)=1, 3!L 3 (x)=−x^3 +9x^2 − 18 x+6,

L 1 (x)=−x+1, 4!L 4 (x)=x^4 − 16 x^3 +72x^2 − 96 x+24,

2!L 2 (x)=x^2 − 4 x+2, 5!L 5 (x)=−x^5 +25x^4 − 200 x^3 + 600x^2 − 600 x+ 120.

The functionsL 0 (x),L 1 (x),L 2 (x)andL 3 (x) are plotted in figure 18.7.