scipy.special.jv#
- scipy.special.jv(v, z, out=None) = <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).- 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() 