HP 48gII Graphing Calculator Manual do Utilizador descarregar pdf (Página 532)

100

101

Page 16-54

Legendre’s equation

An equation of the form (1-x

)⋅(d

y/dx

)-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

mnmnm

)!2()!(!2

)!22(

)1()(

−

∑

⋅

−⋅−⋅⋅

−

⋅−=

.....

)!2()!1(!12

)!22(

)!(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, i.e., P

(x) = 1.0.

1 LEGENDRE, result: ‘X’, i.e., P

(x) = x.

2 LEGENDRE, result: ‘(3*X^2-1)/2’, i.e., P

(x) = (3x

-1)/2.

3 LEGENDRE, result: ‘(5*X^3-3*X)/2’, i.e., P

(x) =(5x

-3x)/2.

4 LEGENDRE, result: ‘(35*X^4-30*X^2+3)/8’, i.e.,

(x) =(35x

-30x

+3)/8.

5 LEGENDRE, result: ‘(63*X^5-70*X^3+15*X)/8’, i.e.,

(x) =(63x

-70x

+15x)/8.

The ODE (1-x

)⋅(d

y/dx

)-2⋅x⋅ (dy/dx)+[n⋅ (n+1)-m

/(1-x

)] ⋅y = 0, has for

solution the function y(x) = P

(x)= (1-x

)

m/2

⋅(d

Pn/dx

). This function is

referred to as an associated Legendre function

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