scipy.special.yv

scipy.special.yv(v, z, out=None) = <ufunc 'yv'>

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

Parameters

varray_like

Order (float).

zarray_like

Argument (float or complex).

outndarray, optional

Optional output array for the function results

Returns

Yscalar or ndarray

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

See also

yve

\(Y_v\) with leading exponential behavior stripped off.

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).

References

1

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