scipy.special.yve#
- scipy.special.yve(v, z, out=None) = <ufunc 'yve'>#
- Exponentially scaled Bessel function of the second kind of real order. - Returns the exponentially scaled Bessel function of the second kind of real order v at complex z: - yve(v, z) = yv(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 results 
 
- Returns:
- Yscalar or ndarray
- Value of the exponentially scaled Bessel function. 
 
 - See also - yv
- Unscaled Bessel function of the second kind of real order. 
 - Notes - 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). - Exponentially scaled Bessel functions are useful for large 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 - yvand- yvefor large complex arguments for z by computing their values for order- v=1at- z=1000j. We see that- yvreturns nan but- yvereturns a finite number:- >>> import numpy as np >>> from scipy.special import yv, yve >>> v = 1 >>> z = 1000j >>> yv(v, z), yve(v, z) ((nan+nanj), (-0.012610930256928629+7.721967686709076e-19j)) - For real arguments for z, - yvereturns the same as- yvup to floating point errors.- >>> v, z = 1, 1000 >>> yv(v, z), yve(v, z) (-0.02478433129235178, -0.02478433129235179) - The function can be evaluated for several orders at the same time by providing a list or NumPy array for v: - >>> yve([1, 2, 3], 1j) array([-0.20791042+0.14096627j, 0.38053618-0.04993878j, 0.00815531-1.66311097j]) - In the same way, the function can be evaluated at several points in one call by providing a list or NumPy array for z: - >>> yve(1, np.array([1j, 2j, 3j])) array([-0.20791042+0.14096627j, -0.21526929+0.01205044j, -0.19682671+0.00127278j]) - It is also possible to evaluate several orders at several points at the same time by providing arrays for v and z with broadcasting compatible shapes. Compute - yvefor two different orders v and three points z resulting in a 2x3 array.- >>> v = np.array([[1], [2]]) >>> z = np.array([3j, 4j, 5j]) >>> v.shape, z.shape ((2, 1), (3,)) - >>> yve(v, z) array([[-1.96826713e-01+1.27277544e-03j, -1.78750840e-01+1.45558819e-04j, -1.63972267e-01+1.73494110e-05j], [1.94960056e-03-1.11782545e-01j, 2.02902325e-04-1.17626501e-01j, 2.27727687e-05-1.17951906e-01j]])