suivant: Tchebychev polynomial of second
monter: Orthogonal polynomials
précédent: Laguerre polynomials: laguerre
Table des matières
Index
Tchebychev polynomials of first kind: tchebyshev1
tchebyshev1 takes as argument an integer n and optionnally a
variable name (by default x).
tchebyshev1 returns the Tchebychev polynomial of first kind
of degree n.
The Tchebychev polynomial of first kind T(n, x) is defined by
T(n, x) = cos(n.arccos(x))
and verify the recurrence relation:
T(0, x) = 1, T(1, x) = x, T(n, x) = 2xT(n - 1, x) - T(n - 2, x)
The polynomials T(n, x) are orthogonal for the scalar product
<
f,
g > =
![$\displaystyle \int_{{-1}}^{{+1}}$](img315.png)
dx
Input :
tchebyshev1(4)
Output :
8*x^
4+-8*x^
2+1
Input :
tchebyshev1(4,y)
Output :
8*y^
4+-8*y^
2+1
Indeed
cos(4.x) |
= |
Re((cos(x) + i.sin(x))4) |
|
|
= |
cos(x)4 -6.cos(x)2.(1 - cos(x)2) + ((1 - cos(x)2)2 |
|
|
= |
T(4, cos(x)) |
|
suivant: Tchebychev polynomial of second
monter: Orthogonal polynomials
précédent: Laguerre polynomials: laguerre
Table des matières
Index
giac documentation written by Renée De Graeve and Bernard Parisse