HP 48GII, Руководство пользователя, Страница: 532 / 864

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.