scipy.special.yn#
- scipy.special.yn(n, x, out=None) = <ufunc 'yn'>#
Bessel function of the second kind of integer order and real argument.
- Parameters:
- narray_like
Order (integer).
- xarray_like
Argument (float).
- outndarray, optional
Optional output array for the function results
- Returns:
- Yscalar or ndarray
Value of the Bessel function, \(Y_n(x)\).
See also
Notes
Wrapper for the Cephes [1] routine
yn
.The function is evaluated by forward recurrence on n, starting with values computed by the Cephes routines
y0
andy1
. If n = 0 or 1, the routine fory0
ory1
is called directly.References
[1]Cephes Mathematical Functions Library, http://www.netlib.org/cephes/
Examples
Evaluate the function of order 0 at one point.
>>> from scipy.special import yn >>> yn(0, 1.) 0.08825696421567697
Evaluate the function at one point for different orders.
>>> yn(0, 1.), yn(1, 1.), yn(2, 1.) (0.08825696421567697, -0.7812128213002888, -1.6506826068162546)
The evaluation for different orders can be carried out in one call by providing a list or NumPy array as argument for the v parameter:
>>> yn([0, 1, 2], 1.) array([ 0.08825696, -0.78121282, -1.65068261])
Evaluate the function at several points for order 0 by providing an array for z.
>>> import numpy as np >>> points = np.array([0.5, 3., 8.]) >>> yn(0, points) array([-0.44451873, 0.37685001, 0.22352149])
If z is an array, the order parameter v must be broadcastable to the correct shape if different orders shall be computed in one call. To calculate the orders 0 and 1 for an 1D array:
>>> orders = np.array([[0], [1]]) >>> orders.shape (2, 1)
>>> yn(orders, points) array([[-0.44451873, 0.37685001, 0.22352149], [-1.47147239, 0.32467442, -0.15806046]])
Plot the functions of order 0 to 3 from 0 to 10.
>>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(0., 10., 1000) >>> for i in range(4): ... ax.plot(x, yn(i, x), label=f'$Y_{i!r}$') >>> ax.set_ylim(-3, 1) >>> ax.legend() >>> plt.show()