Chapter5-2
5.2 Legendre’s Differential Equation
Legendre’s equation arises when solving partial differential equations involving the Laplacian in spherical coordinate. The ODE in the r direction of spherical coordinate usually takes the form
(1 - x2) - 2x + my = 0, - 1 < x < 1 (5.2-1)
Since p(x) = , q(x) = , and r(x) = 0 are analytic at x = 0, the equation can be represented by a power series solution of the form
y = (5.2-2)
Differentiating the series solution (5.2-2) yields
y’ = , y” =
Substituting the function and its derivatives into Eq. (5.2-1) yields
(1 - x2) - 2x + m= 0
- - 2+ m= 0 (5.2-3)
The terms with xm-2 can be changed to xm by replacing m with m + 2
=
Equation (5.2-3) becomes
- - 2x + m= 0
+ 2a2 + ma0 + (6a3 - 2a1 + ma1)x = 0
a2 = a0, a3 = a1
am+2 = am
For m = Þ am+2 = am = - am
a2 = a0 a3 = a1
a4 = a2 = a0
a5 = a3 = a1
The general solution is then
y= a0y1+ a1y2
where
y1= 1 x2 + x2 - ...
y2= x x3 + x5 + ...
If n is an even integer then (when m = n) an+2 = - an = 0 = an+4 = an+6 = ...
If n = 2, y1= 1 x2 = 1 - 3x2
If n = 4, y1= 1 - 10x2 + x4
If n is an even integer then y1is a polynomial of degree n, and y2has the form of an infinite series. Similarly, if n is an odd integer then y2is a polynomial of degree n, and y1is an infinite series.
If n = 1, y2= x
If n = 3, y2= x - x3
If n = 5, y2= x - x3 + x5
For the polynomial solutions, the non-vanishing coefficients can be expressed in terms of the coefficients an of the highest power of x of the polynomial. We need a backward recurrence relation that takes us from am to am-2. From the recurrence relation
am+2 = - am
Replacing m by m – 2 yields
am = - am-2
The backward recurrence relation becomes
am-2 = - am
For m = n,
an-2 = - an
The coefficient an is still arbitrary. It is standard to choose an = so that all the polynomials will have the value unity at x = 1. It we do that, the polynomials are called Legendre polynomials of degree n: Pn(x)
Pn(x) = xn-2m
where M = n/2 if n is even, and M = (n – 1)/2 if n is odd.
The first few Legendre polynomials are
P0(x) = 1 P1(x) = x
P2(x) = (3x2 - 1) P3(x) = (5x3 - 3x)
P4(x) = (35x4 - 30x2 + 3) P5(x) = (63x5 - 70x3 + 15x)
In summary: For n = 0, 1, 2, ... the Legrendre equation of order n
(1 - x2) - 2x + n(n + 1)y = 0, - 1 < x < 1
has two linearly independent solutions y1and y2
y1= 1 x2 + x2 - ...
y2= x x3 + x5 + ...
When n is an even integer y1 is a polynomial Pn(x) of degree n and y2 has the form of an infinite series. Legendre’s function of the second kind is defined as
Qn(x) = y1 y2 n even
When n is an odd integer y2 is a polynomial Pn(x) of degree n and y1 has the form of an infinite series. Legendre’s function of the second kind for this case is defined as
Qn(x) = - y1 y2 n odd
The general solution of Legrendre equation is then
y= C1 Pn(x) + C2 Qn(x)
Legendre function of the second kind converges on the interval - 1 < x < 1 and diverges at the end points. The first few Legendre functions of the second kind are
Q0(x) = Q1(x) = x Q0(x) - 1
Q2(x) = P2(x)Q0(x) - x Q3(x) = P3(x)Q0(x) - x2 +
Q4(x) = P4(x)Q0(x) - x3 + x Q5(x) = P5(x)Q0(x) - x4 + x2 -
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