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 > = 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