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

yv: For real order and real or complex argument.
y0: faster implementation of this function for order 0
y1: faster implementation of this function for order 1

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 and y1. If n = 0 or 1, the routine for y0 or y1 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()