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# scipy.special.jve

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

  Exponentially scaled Bessel function 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).
  Returns: | J : ndarray Value of the exponentially scaled Bessel function.

  Notes

  For positive v values, the computation is carried out using the AMOS [R460] zbesj routine, which exploits the connection to the modified Bessel function \(I_v\),

  \[ \begin{align}\begin{aligned}J_v(z) = \exp(n\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)\\J_v(z) = \exp(-n\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)\end{aligned}\end{align} \]

  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).

  References

  [R460] | (1, 2) Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/