Chebyshev2nd

Chebyshev2nd(self, order: int)

Chebyshev polynomials of the second kind.

Parameters

order : int

The maximum order of the polynomials.

Notes

The (normalised) Chebyshev polynomials of the second kind, defined on \((-1, 1)\), are given by (Boyd 2001) \[ p_{k}(x) = \frac{\sin((k+1)\arccos(x))}{\sin{(\arccos(x))}}, \qquad k = 0, 1, \dots, n. \] The polynomials are orthogonal with respect to the (normalised) weighting function given by \[ \lambda(x) = \frac{2\sqrt{1-x^{2}}}{\pi}. \]

References

Boyd, John P. 2001. Chebyshev and Fourier Spectral Methods. https://link.springer.com/book/9783540514879.