scipy.special.yve — SciPy v1.15.3 Manual (original) (raw)

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 yv and yve for large complex arguments for _z_by computing their values for order v=1 at z=1000j. We see thatyv 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]])