1000 Solved Problems in Modern Physics

1.1 Basic Concepts and Formulae 7

Ln(x)=ex

dn

dxn

(xne−x) (Rodrigue’s formula) (1.43)

The first few polynomials are:

Lo(x)= 1 ,L 1 (x)=−x+ 1 ,L 2 (x)=x^2 − 4 x+ 2

L 3 (x)=−x^3 + 9 x^2 − 18 x+ 6 ,L 4 (x)=x^4 − 16 x^3 + 72 x^2 − 96 x+24 (1.44)

Generating function:

e−xs/(1−s)

1 −s

=

∑∞

n= 0

Ln(x)sn

n!

(1.45)

Recurrence formulas:

Ln+ 1 (x)−(2n+ 1 −x)Ln(x)+n^2 Ln− 1 (x)= 0

xL′n(x)=nLn(x)−n^2 Ln− 1 (x) (1.46)

Orthonormal properties:
∫∞

0

e−xLm(x)Ln(x)dx= 0 m

=n (1.47)

∫∞

0

e−x{Ln(x)}^2 dx=(n!)^2 (1.48)

Bessel functions:(Jn(x))

Differential equation of ordern

x^2 y′′+xy′+(x^2 −n^2 )y= 0 n≥ 0 (1.49)

Expansion formula:

Jn(x)=

∑∞

k= 0

(−1)k(x/2)^2 k−n

k!Γ(k+ 1 −n)

(1.50)

Properties:

J−n(x)=(−1)nJn(x) n= 0 , 1 , 2 ,... (1.51)

Jo′(x)=−J 1 (x) (1.52)

Jn+ 1 (x)=

2 n

x

Jn(x)−Jn− 1 (x) (1.53)

Generating function:

ex(s−^1 /s)/^2 =

∑∞

n=−∞

Jn(x)tn (1.54)