The Chemistry Maths Book, Second Edition

13.4 The Legendre equation 377

interest in the physical sciences.

3

By convention, the nonzero arbitrary constant in

(13.16) is chosen so thatP

l

(1) 1 = 11. Then, forl 1 = 1 0, 1, 2, 3, =,

(13.19)

and the series is to be continued down to the constant term. The first few of these

polynomials are

P

0

(x) 1 = 11 P

1

(x) 1 = 1 x

(13.20)

EXAMPLE 13.5Show that the polynomial P

3

(x)is a solution of the Legendre

equation (13.13) forl 1 = 13.

We have

Therefore, forl 1 = 13 ,

(1 1 − 1 x

2

)y′′ 1 − 12 xy′ 1 + 1 l(l 1 + 1 1)y

= 1 (− 151 − 1151 + 1 30)x

3

1 + 1 (15 1 + 131 − 1 18)x 1 = 10

0 Exercise 16

=− × − ×() ( )−+××( )1152−

3

2

5134

1

2

53

223

xxxx xx

yPx x x y x== −,′=−,y x′′=

3

32

1

2

53

3

2

() ( ) ( 51 ) 15

Px x x x

5

53

1

8

() (=−+63 70 15 )

Px x x

4

42

1

8

() (=−+35 30 3)

Px x x

3

3

1

2

() (=− 53 )

Px x

2

2

1

2

() (=− 31 )

×−

−

−

• 
  

−− −

−

x

ll

l

x

ll l l

ll

()

()

()( )()

(

1

22 1

123

242

2

· lll

x

l

−−

−





















−

12 3

4

)( )



Px

l

l

l

()

()

=

−

!

135 2 1··

3

Adrien-Marie Legendre (1752–1833) has the doubtful distinction of the following footnote in E. T. Bell’s Men

of mathematics: ‘Considerations of space preclude an account of his life; much of his best work was absorbed or

circumvented by younger mathematicians’. The polynomials appeared in Legendre’s Recherches sur l’attraction des

sphéroïdes homogènesof 1785 as the coefficients in the expansion of the potential function(1 1 − 12 h 1 cos 1 θ 1 + 1 h

2

)

− 122

in powers of h. His textbook Éléments géométrieof 1794 did much to reform the teaching of geometry (till then

based on Euclid), and an American edition, Davies’ Legendre(1851), was influential in the United States.