# scipy.special.yve

scipy.special.yve(v, z) = <ufunc 'yve'>

  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).

  Returns
    Y ndarray

      Value of the exponentially scaled Bessel function.

  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/

scipy.special.yv scipy.special.kn