mars.tensor.special.yv#

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

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

Parameters

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

Returns

Y – Value of the Bessel function of the second kind, \(Y_v(x)\).

Return type

ndarray

Notes

For positive v values, the computation is carried out using the AMOS 1 zbesy routine, which exploits the connection to the Hankel Bessel functions \(H_v^{(1)}\) and \(H_v^{(2)}\),

\[Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}).\]

For negative v values the formula,

\[Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v)\]

is used, where \(J_v(z)\) is the Bessel function of the first kind, computed using the AMOS routine zbesj. 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).