Hp 50g Graphing Calculator Manual do Utilizador descarregar pdf (Página 529)

100

101

Page 16-52

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

Bessel’s equation

The ordinary differential equation x

⋅(d

y/dx

) + x⋅ (dy/dx)+ (x

-ν

) ⋅y = 0,

where the parameter ν is a nonnegative real number, is known as Bessel’s

differential equation. Solutions to Bessel’s equation are given in terms of

Bessel functions of the first kind of order

ν:

mnmnm

)!2()!(!2

)!22(

)1()(

−

∑

⋅

−⋅−⋅⋅

−

⋅−=

.....

)!2()!1(!12

)!22(

)!(2

)!2(

−+⋅

−−⋅⋅

−

−⋅

⋅

−n

∑

∞

++Γ⋅⋅

⋅−

⋅=

)1(!2

)1(

)(

xxJ

1 2 ... 524 525 526 527 528 529 530 531 532 533 534 ... 886 887

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