6.30.5 Tchebychev polynomial of the second kind: tchebyshev2
The Tchebychev 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)=sin(x)*U(n,cos(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 Tchebychev polynomials of
the first kind.
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tchebyshev2 takes one mandatory argument and one
optional argument:
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n, an integer.
- Optionally x, a variable name (by default x).
- tchebyshev2(n ⟨,x⟩) returns
the Tchebychev polynomial of second kind of degree n.
Examples.
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Input:
tchebyshev2(3)
Output:
- Input:
tchebyshev2(3,y)
Output:
Indeed:
| sin(4 x)=sin(x)*(8*cos(x)3−4 cos(x))=sin(x)*U(3,cos(x))
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