# scipy.special.jv

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

  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 ndarray

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

  See also

  `jve`

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

  `spherical_jn`

    spherical Bessel functions.

  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`).

  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.ellipeinc scipy.special.jve