6.30.4 Tchebychev polynomials of the first kind: tchebyshev1
The Tchebychev polynomial of first kind T(n,x) is defined by
and satisfy 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
The tchebyshev1 command finds the Tchebychev polynomials of
the first kind.
-
tchebyshev1 takes one mandatory argument and one
optional argument:
-
n, an integer.
- Optionally x, a variable name (by default x).
- tchebyshev1(n ⟨,x⟩) returns
the Tchebychev polynomial of first kind of degree n.
Examples.
-
Input:
tchebyshev1(4)
Output:
- Input:
tchebyshev1(4,y)
Output:
Indeed
| cos(4x) | = | Re((cos(x)+i sin(x))4) |
| | = | cos(x)4−6cos(x)2 (1−cos(x)2)+((1−cos(x)2)2 |
| | = | T(4,cos(x))
|
|