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# scipy.special.jve

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

  Exponentially scaled Bessel function of the first kind of order v.

  Defined as:

      jve(v, z) = jv(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 values

  Returns:
    J scalar or ndarray

      Value of the exponentially scaled Bessel function.

  See also

  `jv`

    Unscaled Bessel function of the first kind

  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).

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

  References

  [1]

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

  Examples

  Compare the output of `jv` and `jve` for large complex arguments for z by computing their values for order `v=1` at `z=1000j`. We see that `jv` overflows but `jve` returns a finite number:

      >>> import numpy as np
      >>> from scipy.special import jv, jve
      >>> v = 1
      >>> z = 1000j
      >>> jv(v, z), jve(v, z)
      ((inf+infj), (7.721967686709077e-19+0.012610930256928629j))

  For real arguments for z, `jve` returns the same as `jv`.

      >>> v, z = 1, 1000
      >>> jv(v, z), jve(v, z)
      (0.004728311907089523, 0.004728311907089523)

  The function can be evaluated for several orders at the same time by providing a list or NumPy array for v:

      >>> jve([1, 3, 5], 1j)
      array([1.27304208e-17+2.07910415e-01j, -4.99352086e-19-8.15530777e-03j,
             6.11480940e-21+9.98657141e-05j])

  In the same way, the function can be evaluated at several points in one call by providing a list or NumPy array for z:

      >>> jve(1, np.array([1j, 2j, 3j]))
      array([1.27308412e-17+0.20791042j, 1.31814423e-17+0.21526929j,
             1.20521602e-17+0.19682671j])

  It is also possible to evaluate several orders at several points at the same time by providing arrays for v and z with compatible shapes for broadcasting. Compute `jve` for two different orders v and three points z resulting in a 2x3 array.

      >>> v = np.array([[1], [3]])
      >>> z = np.array([1j, 2j, 3j])
      >>> v.shape, z.shape
      ((2, 1), (3,))

      >>> jve(v, z)
      array([[1.27304208e-17+0.20791042j,  1.31810070e-17+0.21526929j,
              1.20517622e-17+0.19682671j],
             [-4.99352086e-19-0.00815531j, -1.76289571e-18-0.02879122j,
              -2.92578784e-18-0.04778332j]])