scipy.special.jve#
- scipy.special.jve(v, z, out=None) = <ufunc 'jve'>#
- Exponentially scaled Bessel function of the first kind of order v. - Defined as: - jve(v, z) = jv(v, z) * exp(-abs(z.imag)) - 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 exponentially scaled Bessel function. 
 
 - See also - jv
- Unscaled Bessel function of the first kind 
 - 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). - Exponentially scaled Bessel functions are useful for large arguments z: for these, the unscaled Bessel functions can easily under-or overflow. - References [1]- Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/ - Examples - Compare the output of - jvand- jvefor large complex arguments for z by computing their values for order- v=1at- z=1000j. We see that- jvoverflows but- jvereturns a finite number:- >>> import numpy as np >>> from scipy.special import jv, jve >>> v = 1 >>> z = 1000j >>> jv(v, z), jve(v, z) ((inf+infj), (7.721967686709077e-19+0.012610930256928629j)) - For real arguments for z, - jvereturns the same as- jv.- >>> v, z = 1, 1000 >>> jv(v, z), jve(v, z) (0.004728311907089523, 0.004728311907089523) - The function can be evaluated for several orders at the same time by providing a list or NumPy array for v: - >>> jve([1, 3, 5], 1j) array([1.27304208e-17+2.07910415e-01j, -4.99352086e-19-8.15530777e-03j, 6.11480940e-21+9.98657141e-05j]) - In the same way, the function can be evaluated at several points in one call by providing a list or NumPy array for z: - >>> jve(1, np.array([1j, 2j, 3j])) array([1.27308412e-17+0.20791042j, 1.31814423e-17+0.21526929j, 1.20521602e-17+0.19682671j]) - It is also possible to evaluate several orders at several points at the same time by providing arrays for v and z with compatible shapes for broadcasting. Compute - jvefor two different orders v and three points z resulting in a 2x3 array.- >>> v = np.array([[1], [3]]) >>> z = np.array([1j, 2j, 3j]) >>> v.shape, z.shape ((2, 1), (3,)) - >>> jve(v, z) array([[1.27304208e-17+0.20791042j, 1.31810070e-17+0.21526929j, 1.20517622e-17+0.19682671j], [-4.99352086e-19-0.00815531j, -1.76289571e-18-0.02879122j, -2.92578784e-18-0.04778332j]])