Mathematical Tools for Physics

4—Differential Equations 108

a 2 =a 0

s(s+ 1)−C

(s+ 2)(s+ 1)

, then a 4 =a 2

(s+ 2)(s+ 3)−C

(s+ 4)(s+ 3)

, etc. (38)

This looks messier than it is. Notice that the only combination of indices that shows up isn+s. The indexsis
0 or 1, andnis an even number, son+scovers the non-negative integers: 0, 1, 2,...
The two solutions to the Legendre differential equation come from the two cases,s= 0, 1.

s= 0 : a 0

[

1 +

(

−C

2

)

x^2 +

(

−C

2

)(

2. 3 −C

4. 3

)

x^4 +

(

−C

2

)(

2. 3 −C

4. 3

)(

4. 5 −C

6. 5

)

x^6 ···

]

s= 1 : a′ 0

[

x+

(

1. 2 −C

3. 2

)

x^3 +

(

1. 2 −C

3. 2

)(

3. 4 −C

5. 4

)

x^5 +···

] (39)

and the general solution is a sum of these.
This procedure gives both solutions to the differential equation, one with even powers and one with odd
powers. Both are infinite series and are called Legendre Functions. An important point about both of them is
that they blow up asx→ 1. This fact shouldn’t be too surprising, because the differential equation ( 36 ) has a
singular point there.

y′′−

2 x

(1 +x)(1−x)

y′+

C

(1 +x)(1−x)

y= 0

It’s a regular singular point, but it is still singular. A detailed calculation shows that these solutions behave as
ln(1−x).
There is an exception! If the constantCis for exampleC= 6, then withs= 0the equations ( 38 ) are

a 2 =a 0

− 6

2

, a 4 =a 2

6 − 6

12

= 0, a 6 =a 8 =...= 0

The infinite series terminates in a polynomial

a 0 +a 2 x^2 =a 0 [1− 3 x^2 ]

This (after a conventional rearrangement) is a Legendre Polynomial,

P 2 (x) =

3

2

x^2 −

1

2