HP 50g User Manual Page 529

Graphing calculator.

P
(
n
2 (
n
2
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.,
5 LEGENDRE, result: '(63*X^5-70*X^3+15*X)/8', i.e.,
2
The ODE (1-x
solution the function y(x) = P
referred to as an associated Legendre function.
Bessel's equation
The ordinary differential equation x
where the parameter
differential equation.
Bessel functions of the first kind of order :
M
m
x
)
(
) 1
m
0
n
)!
n
x
2
(
n
) !
2
P
(x) =(35x
4
P
(x) =(63x
5
2
2
) (d
y/dx
)-2 x (dy/dx)+[n (n+1)-m
m
(x)= (1-x
n
is a nonnegative real number, is known as Bessel's
Solutions to Bessel's equation are given in terms of
J
(
x
)
x
2 (
n
2
n
2
m
( !
n
m
2 (
n
2
)!
n
1
( !
n
1
( )!
n
4
2
-30x
+3)/8.
5
3
-70x
+15x)/8.
2
m/2
m
)
(d
2
2
2
(d
y/dx
) + x (dy/dx)+ (x
m
(
) 1
2
m
2
m
!
m
0
m
)!
n
x
)!
(
n
2
m
)!
n
2
x
...
2
)!
i.e.,
P
(x) = 1.0.
0
i.e.,
P
(x) = x.
1
i.e.,
P
(x) = (3x
2
i.e.,
P
(x) =(5x
3
2
2
/(1-x
)] y = 0, has for
m
Pn/dx
). This function is
2
m
x
,
(
m
) 1
2
m
..
2
-1)/2.
3
-3x)/2.
2
2
-
) y = 0,
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