11.4.5 Chebyshev polynomial of the second kind
The Chebyshev polynomial of second kind U(n,x) is defined by:
| U(n,x)= | | sin((n+1) arccos x) |
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| sin(arccos x) |
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or equivalently:
| sin((n+1)x)=U(n,cos x)sin x.
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These satisfy the recurrence relation:
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| | U(0,x) | =1 | | | | | | | | | |
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U(1,x) | =2x | | | | | | | | | |
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U(n,x) | =2xU(n−1,x)−U(n−2,x)
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The polynomials U(n,x) are orthogonal for the scalar product
The tchebyshev2
command finds the Chebyshev polynomials of
the first kind.
- tchebyshev2 takes one mandatory argument and one
optional argument:
- n, an integer.
- Optionally x, a variable name (by default x).
- tchebyshev2(n ⟨,x⟩) returns
the Chebyshev polynomial of second kind of degree n.
Examples
Indeed, sin(4 x)=sin(x) (8 cos(x)3−4 cos(x))=sin(x) U(3,cos(x)).