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The LEGENDRE function
A Legendre polynomial of order n is a polynomial function that solves the
differential equation
0)1(2)1(
2
2
2
=⋅+⋅+⋅⋅−⋅− ynn
dx
dy
x
dx
yd
x
To obtain the n-th order Legendre polynomial, use LEGENDRE(n), e.g.,
LEGENDRE(3) = ‘(5*X^3-3*X)/2’
LEGENDRE(5) = ‘(63*X ^5-70*X^3+15*X)/8’
The PCOEF function
Given an array containing the roots of a polynomial, the function PCOEF
generates an array containing the coefficients of the corresponding
polynomial. The coefficients correspond to decreasing order of the
independent variable. For example: PCOEF([-2,–1,0,1,1,2]) = [1. –1. –5. 5.
4. –4. 0.], which represents the polynomial X
6
-X
5
-5X
4
+5X
3
+4X
2
-4X.
The PROOT function
Given an array containing the coefficients of a polynomial, in decreasing
order, the function PROOT provides the roots of the polynomial. Example,
from X
2
+5X-6 =0, PROOT([1, –5, 6]) = [2. 3.].
The PTAYL function
Given a polynomial P(X) and a number a, the function PTAYL is used to obtain
an expression Q(X-a) = P(X), i.e., to develop a polynomial in powers of (X- a).
This is also known as a Taylor polynomial, from which the name of the
function, Polynomial & TAYLor, follow:
For example, PTAYL(‘X^3-2*X+2’,2) = ‘X^3+6*X^2+10*X+6’.
In actuality, you should interpret this result to mean
‘(X-2) ^3+6*(X-2) ^2+10*(X-2) +6’.
Let’s check by using the substitution: ‘X = x – 2’. We recover the original
polynomial, but in terms of lower-case x rather than upper-case x.
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