The Chemistry Maths Book, Second Edition

13.4 The Legendre equation 375

This is zero if the coefficient of each power of xis separately zero. The coefficient of

x

1

is 2 a

1

so thata

1

1 = 10 and it follows thata

m

1 = 10 for odd values of m. For the even

values, as in Example 13.2,

Therefore,

= 1 a

0

x

− 122

1 sin 1 x

By convention and the function is the Bessel function

of ordern 1 = 1122. The same procedure withr 1 = 1 − 122 gives the Bessel function of

order− 122 (Exercise 13).

0 Exercises 13–15

13.4 The Legendre equation

The Legendre equation is

(1 1 − 1 x

2

)y′′ 1 − 12 xy′ 1 + 1 l(l 1 + 1 1)y 1 = 10 (13.13)

where lis a real number. This equation arises whenever a physical problem in three

dimensions is formulated in terms of spherical polar coordinates, r, θand φ(see

Section 10.2), in which case the variable xis replaced by cos 1 θ. Of interest in the

physical sciences are then those solutions that are finite in the interval− 11 ≤ 1 x 1 ≤ 1 + 1

(corresponding to 01 ≤ 1 θ 1 ≤ 1 π).

Equation (13.13) can be solved by the series method in the way described in Section

13.2. The series

(13.14)

and its derivativesy′andy′′are substituted into (13.13). Then, setting the coefficient

of each power of xequal to zero gives the recurrence relation

(13.15)

For the even values of m,

a

ll

aa

ll

a

lll

204 2

1

2

23

34

2

=−

• 
  

=−

−+

×

=+

() −

,

()() ()(+++

!

13

4

0

)( )

,

l

a ...

a

mm ll

mm

a

lmlm

mm+

=

+− +

++

=−

−++

2

11

12

()() 1

()( )

()( ))

()( )mm

a

m

++ 12

yaxaaxax

m

m

m

==+++

∑

01 2

2



Jx

x

x

12

2

()= sin

π

a

0

= 2 π

yx ax

xxx

() ax

!!!

= −+−+





















=

−

0

12

246

0

1

357



112

357

357

x

xxx

−+− +





















!!!



aaa aaa a

204 206 4

1

3

1

54

1

5

1

76

1

7

=− , =−

×

=+

!

,=−

×

=−

!!

aa

0

,