HP 48gII - Manual (Page 532)

Page 16-54
Legendre’s equation
An equation of the form (1-x 2 ) ⋅(d 2 y/dx 2 )-2 ⋅x⋅ (dy/dx)+n⋅ (n+1) ⋅y = 0, where n
is a real number, is known as the Legendre’s differential equation. Any
solution for this equation is known as a Legendre’s function. When n is a
nonnegative integer, the solutions are called Legendre’s polynomials.
Legendre’s polynomial of order n is given by
m n
M
m
n
m
n
x
m n m n m
m n
x P 2
0
)! 2 ( )! ( ! 2
)! 2 2 (
) 1 ( ) ( −
=
∑
⋅
− ⋅ − ⋅ ⋅
−
⋅ − =
.. ...
)! 2 ( )! 1 ( ! 1 2
)! 2 2 (
) ! ( 2
)! 2 (
2
2
− + ⋅
− − ⋅ ⋅
−
− ⋅
⋅
= − n
n
n
n
x
n n
n
x
n
n
where M = n/2 or (n-1)/2, whichever is an integer.
Legendre’s polynomials are pre-programmed in the calculator and can be
recalled by using the function LEGENDRE given the order of the polynomial, n.
The function LEGENDRE can be obtained from the command catalog
(
‚N) or through the menu ARITHMETIC/POLYNOMIAL menu (see
Chapter 5). In RPN mode, the first six Legendre polynomials are obtained as
follows:
0 LEGENDRE, result: 1,
i.e., P
0
(x) = 1.0.
1 LEGENDRE, result: ‘X’, i.e., P
1
(x) = x.
2 LEGENDRE, result: ‘(3*X^2-1)/2’, i.e., P
2
(x) = (3x 2 -1)/2.
3 LEGENDRE, result: ‘(5*X^3-3*X)/2’, i.e., P
3
(x) =(5x 3 -3x)/2.
4 LEGENDRE, result: ‘(35*X^4-30*X^2+3)/8’, i.e.,
P
4
(x) =(35x 4 -30x 2 +3)/8.
5 LEGENDRE, result: ‘(63*X^5-70*X^3+15*X)/8’, i.e.,
P
5
(x) =(63x 5 -70x 3 +15x)/8.
The ODE (1-x
2 ) ⋅(d 2 y/dx 2 )-2 ⋅x⋅ (dy/dx)+[n⋅ (n+1)-m 2 /(1-x 2 )] ⋅y = 0, has for
solution the function y(x) = P
n
m (x)= (1-x 2 ) m/2 ⋅(d m Pn/dx m ). This function is
referred to as an associated Legendre function.