scipy.special.jve#

scipy.special.jve(v, z, out=None) = <wrapped_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.

Array API Standard Support

jve has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. jve does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

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 jv and jve for large complex arguments for z by computing their values for order v=1 at z=1000j. We see that jv overflows but jve returns 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, jve returns 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 jve for 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]])