Hewlett-Packard F2226A - 48GII Graphic Calculator User's Manual
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Brand: HP
Category: Calculator
Type: Operation & user’s manual
Model: HP 48gII , HP F2226A - 48GII Graphic Calculator
Pages: 864 (9.97 Mb)
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
mn
M
m
n
m
n
x
mnmnm
mn
xP
2
0
)!2()!(!2
)!22(
)1()(
−
=
∑
⋅
−⋅−⋅⋅
−
⋅−=
.....
)!2()!1(!12
)!22(
)!(2
)!2(
2
2
−+⋅
−−⋅⋅
−
−⋅
⋅
=
−n
n
n
n
x
nn
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
.