tchebyshev2 takes as argument an integer n and optionnally
a variable name (by default x).
tchebyshev2 returns the Tchebychev polynomial of second kind
of degree n.
The Tchebychev polynomial of second kind U(n,x) is defined by:
U(n,x)= |
|
or equivalently:
sin((n+1)x)=sin(x)*U(n,cos(x)) |
The U(n,x) verifies the recurrence relation:
U(0,x)=1, U(1,x)=2x, U(n,x)=2xU(n−1,x)−U(n−2,x) |
The polynomials U(n,x) are orthogonal for the scalar product
<f,g>= | ∫ |
| f(x)g(x) | √ |
| dx |
Input :
Output :
^
3+-4*xInput :
Output :
^
3+-4*yIndeed:
sin(4.x)=sin(x)*(8*cos(x)3−4.cos(x))=sin(x)*U(3,cos(x)) |