scipy.special.eval_chebyu#

scipy.special.eval_chebyu(n, x, out=None) = <ufunc 'eval_chebyu'>#

Evaluate Chebyshev polynomial of the second kind at a point.

The Chebyshev polynomials of the second kind can be defined via the Gauss hypergeometric function \({}_2F_1\) as

\[U_n(x) = (n + 1) {}_2F_1(-n, n + 2; 3/2; (1 - x)/2).\]

When \(n\) is an integer the result is a polynomial of degree \(n\). See 22.5.48 in [AS] (or equivalently [DLMF]) for details.

Parameters:

narray_like: Degree of the polynomial. If not an integer, the result is determined via the relation to the Gauss hypergeometric function.
xarray_like: Points at which to evaluate the Chebyshev polynomial
outndarray, optional: Optional output array for the function values

Returns:

Uscalar or ndarray: Values of the Chebyshev polynomial

See also

roots_chebyu: roots and quadrature weights of Chebyshev polynomials of the second kind
chebyu: Chebyshev polynomial object
eval_chebyt: evaluate Chebyshev polynomials of the first kind
hyp2f1: Gauss hypergeometric function

References

[AS]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

[DLMF]

NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/18.5.E11_4

Examples

>>> import numpy as np
>>> from scipy.special import eval_chebyu
>>> import matplotlib.pyplot as plt

Evaluate the order 3 polynomial at a few values of x.

>>> eval_chebyu(3, [0, 0.25, 0.5])
array([-0.   , -0.875, -1.   ])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(-1, 1, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_chebyu(n, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Chebyshev Polynomials of the Second Kind")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$U_n(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

../../_images/scipy-special-eval_chebyu-1.png