mars.tensor.special.jve

mars.tensor.special.jve(v, z, **kwargs)

Exponentially scaled Bessel function 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).

Returns

J – Value of the exponentially scaled Bessel function.

Return type

ndarray

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).

References

1

Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/