mars.tensor.special.jv#

mars.tensor.special.jv(v, z, **kwargs)[source]#

Bessel function of the first kind of real order and complex argument.

Parameters

v (array_like) – Order (float).
z (array_like) – Argument (float or complex).

Returns

J – Value of the Bessel function, \(J_v(z)\).

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).

Not to be confused with the spherical Bessel functions (see spherical_jn).