scipy.special.jv#

scipy.special.jv(v, z, out=None) = <wrapped_ufunc 'jv'>#

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

Parameters:

varray_like: Order (float).
zarray_like: Argument (float or complex).
outndarray, optional: Optional output array for the function values

Returns:

Jscalar or ndarray: Value of the Bessel function, \(J_v(z)\).

See also

jve: \(J_v\) with leading exponential behavior stripped off.
spherical_jn: spherical Bessel functions.
j0: faster version of this function for order 0.
j1: faster version of this function for order 1.

Notes

For positive v values, the computation is carried out using the AMOS [1] zbesj routine, which exploits the connection to the modified Bessel function \(I_v\),

\[ \begin{align}\begin{aligned}J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)\\J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)\end{aligned}\end{align} \]

For negative v values the formula,

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

is used, where \(Y_v(z)\) is the Bessel function of the second kind, computed using the AMOS routine zbesy. 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).

Not to be confused with the spherical Bessel functions (see spherical_jn).

Array API Standard Support

jv has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. jv does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[1]

Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/

Examples

Evaluate the function of order 0 at one point.

>>> from scipy.special import jv
>>> jv(0, 1.)
0.7651976865579666

Evaluate the function at one point for different orders.

>>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)
(0.7651976865579666, 0.44005058574493355, 0.24029783912342725)

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:

>>> jv([0, 1, 1.5], 1.)
array([0.76519769, 0.44005059, 0.24029784])

Evaluate the function at several points for order 0 by providing an array for z.

>>> import numpy as np
>>> points = np.array([-2., 0., 3.])
>>> jv(0, points)
array([ 0.22389078,  1.        , -0.26005195])

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 a 1D array:

>>> orders = np.array([[0], [1]])
>>> orders.shape
(2, 1)

>>> jv(orders, points)
array([[ 0.22389078,  1.        , -0.26005195],
       [-0.57672481,  0.        ,  0.33905896]])

Plot the functions of order 0 to 3 from -10 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> for i in range(4):
...     ax.plot(x, jv(i, x), label=f'$J_{i!r}$')
>>> ax.legend()
>>> plt.show()