maxframe.tensor.special.yve#

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

Exponentially scaled Bessel function of the second kind of real order.

Returns the exponentially scaled Bessel function of the second kind of real order v at complex z:

yve(v, z) = yv(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 results

Returns:

Y – Value of the exponentially scaled Bessel function.

Return type:

scalar or ndarray

See also

yv: Unscaled Bessel function of the second kind of real order.

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).

Exponentially scaled Bessel functions are useful for large z: for these, the unscaled Bessel functions can easily under-or overflow.

References