scipy.special.yn — SciPy v1.15.2 Manual (original) (raw)
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
For real order and real or complex argument.
faster implementation of this function for order 0
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
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()
