scipy.special.yv(v, z, out=None) = <ufunc 'yv'>#

Bessel function of the second kind of real order and complex argument.


Order (float).


Argument (float or complex).

outndarray, optional

Optional output array for the function results

Yscalar or ndarray

Value of the Bessel function of the second kind, \(Y_v(x)\).

See also


\(Y_v\) with leading exponential behavior stripped off.


faster implementation of this function for order 0


faster implementation of this function for order 1


For positive v values, the computation is carried out using the AMOS [1] zbesy routine, which exploits the connection to the Hankel Bessel functions \(H_v^{(1)}\) and \(H_v^{(2)}\),

\[Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}).\]

For negative v values the formula,

\[Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v)\]

is used, where \(J_v(z)\) is the Bessel function of the first kind, computed using the AMOS routine zbesj. Note that the second term is exactly zero for integer v; to improve accuracy the second term is explicitly omitted for v values such that v = floor(v).



Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”,


Evaluate the function of order 0 at one point.

>>> from scipy.special import yv
>>> yv(0, 1.)

Evaluate the function at one point for different orders.

>>> yv(0, 1.), yv(1, 1.), yv(1.5, 1.)
(0.088256964215677, -0.7812128213002889, -1.102495575160179)

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:

>>> yv([0, 1, 1.5], 1.)
array([ 0.08825696, -0.78121282, -1.10249558])

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.])
>>> yv(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)
>>> yv(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, yv(i, x), label=f'$Y_{i!r}$')
>>> ax.set_ylim(-3, 1)
>>> ax.legend()