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 `yv` and `yve` for large complex arguments for `z` by computing their values for order ``v=1`` at ``z=1000j``. We see that `yv` returns nan but `yve` returns 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`, `yve` returns the same as `yv` up 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 `yve` for 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]]) Ú