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))

Order (float).


Argument (float or complex).

outndarray, optional

Optional output array for the function values

Jscalar or ndarray

Value of the exponentially scaled Bessel function.

See also


Unscaled Bessel function of the first kind


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.



Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”,


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,

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,

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,
       [-4.99352086e-19-0.00815531j, -1.76289571e-18-0.02879122j,