maxframe.tensor.special.jve#

maxframe.tensor.special.jve(v, z, out=None)[source]#

Exponentially scaled Bessel function of the first kind of order v.

Defined as:

jve(v, z) = jv(v, z) * exp(-abs(z.imag))

Parameters:

v (array_like) – Order (float).
z (array_like) – Argument (float or complex).
out (ndarray, optional) – Optional output array for the function values

Returns:

J – Value of the exponentially scaled Bessel function.

Return type:

scalar or ndarray

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.

References