scipy.special.eval_chebyc#

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

Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a point.

These polynomials are defined as

\[C_n(x) = 2 T_n(x/2)\]

where \(T_n\) is a Chebyshev polynomial of the first kind. See 22.5.11 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 eval_chebyt.
xarray_like: Points at which to evaluate the Chebyshev polynomial
outndarray, optional: Optional output array for the function values

Returns:

Cscalar or ndarray: Values of the Chebyshev polynomial

See also

roots_chebyc: roots and quadrature weights of Chebyshev polynomials of the first kind on [-2, 2]
chebyc: Chebyshev polynomial object
numpy.polynomial.chebyshev.Chebyshev: Chebyshev series
eval_chebyt: evaluate Chebycshev polynomials of the first kind

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/18.1.E3

Examples

>>> import numpy as np
>>> from scipy.special import eval_chebyc, eval_chebyt
>>> import matplotlib.pyplot as plt

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

>>> eval_chebyc(3, [0, 0.25, 0.5])
array([-0.      , -0.734375, -1.375   ])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(-2, 2, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_chebyc(n, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Chebyshev Polynomials of the First Kind on [-2, 2]")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$C_n(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

../../_images/scipy-special-eval_chebyc-1_00_00.png

They are a scaled version of the Chebyshev polynomials of the first kind.

>>> x = np.linspace(-2, 2, 6)
>>> eval_chebyc(3, x)
array([-2.   ,  1.872,  1.136, -1.136, -1.872,  2.   ])
>>> 2 * eval_chebyt(3, x / 2)
array([-2.   ,  1.872,  1.136, -1.136, -1.872,  2.   ])