# Legendre's Equation - HP 48gII User Manual

Graphing calculator.

## Legendre's equation

An equation of the form (1-x
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
P
(
x
)
n
m
=
2 (
n
)!
n
2
2
(
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,
1 LEGENDRE, result: 'X',
2 LEGENDRE, result: '(3*X^2-1)/2',
3 LEGENDRE, result: '(5*X^3-3*X)/2',
4 LEGENDRE, result: '(35*X^4-30*X^2+3)/8', i.e.,
P
5 LEGENDRE, result: '(63*X^5-70*X^3+15*X)/8', i.e.,
P
2
2
The ODE (1-x
)⋅(d
y/dx
solution the function y(x) = P
referred to as an associated Legendre function.
2
2
2
)-2⋅x⋅ (dy/dx)+n⋅ (n+1) ⋅y = 0, where n
)⋅(d
y/dx
2 (
n
2
m
(
) 1
n
2
m
( !
n
m
0
2 (
n
2
)!
n
x
n
2
1
( !
n
1
( )!
n
4
2
(x) =(35x
-30x
+3)/8.
4
5
3
(x) =(63x
-70x
+15x)/8.
5
2
)-2⋅x⋅ (dy/dx)+[n⋅ (n+1)-m
m
2
m/2
⋅(d
m
(x)= (1-x
)
Pn/dx
n
m
)!
n
2
m
x
)!
(
n
2
m
)!
n
2
x
...
..
2
)!
i.e.,
P
(x) = 1.0.
0
i.e.,
P
(x) = x.
1
2
i.e.,
P
(x) = (3x
-1)/2.
2
3
i.e.,
P
(x) =(5x
-3x)/2.
3
2
2
)] ⋅y = 0, has for
/(1-x
m
). This function is
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