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Module « scipy.special »
Signature de la fonction jv
def jv(*args, **kwargs)
Description
help(scipy.special.jv)
jv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])
jv(v, z, out=None)
Bessel function of the first kind of real order and complex argument.
Parameters
----------
v : array_like
Order (float).
z : array_like
Argument (float or complex).
out : ndarray, optional
Optional output array for the function values
Returns
-------
J : scalar or ndarray
Value of the Bessel function, :math:`J_v(z)`.
See Also
--------
jve : :math:`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 :math:`I_v`,
.. math::
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)
For negative `v` values the formula,
.. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v)
is used, where :math:`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`).
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 an 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()
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