# scipy.special.modstruve

scipy.special.modstruve(v, x, out=None) = <ufunc 'modstruve'>

  Modified Struve function.

  Return the value of the modified Struve function of order v at x. The modified Struve function is defined as,

  \[L_v(x) = -\imath \exp(-\pi\imath v/2) H_v(\imath x),\]

  where \(H_v\) is the Struve function.

  Parameters:
    v array_like

      Order of the modified Struve function (float).

    x array_like

      Argument of the Struve function (float; must be positive unless v is an integer).

    out ndarray, optional

      Optional output array for the function results

  Returns:
    L scalar or ndarray

      Value of the modified Struve function of order v at x.

  See also

  `struve`

  Notes

  Three methods discussed in [1] are used to evaluate the function:

  - power series

  - expansion in Bessel functions (if \(|x| < |v| + 20\))

  - asymptotic large-x expansion (if \(x \geq 0.7v + 12\))

  Rounding errors are estimated based on the largest terms in the sums, and the result associated with the smallest error is returned.

  References

  [1]

  NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/11

  Examples

  Calculate the modified Struve function of order 1 at 2.

      >>> import numpy as np
      >>> from scipy.special import modstruve
      >>> import matplotlib.pyplot as plt
      >>> modstruve(1, 2.)
      1.102759787367716

  Calculate the modified Struve function at 2 for orders 1, 2 and 3 by providing a list for the order parameter v.

      >>> modstruve([1, 2, 3], 2.)
      array([1.10275979, 0.41026079, 0.11247294])

  Calculate the modified Struve function of order 1 for several points by providing an array for x.

      >>> points = np.array([2., 5., 8.])
      >>> modstruve(1, points)
      array([  1.10275979,  23.72821578, 399.24709139])

  Compute the modified Struve function for several orders at several points by providing arrays for v and z. The arrays have to be broadcastable to the correct shapes.

      >>> orders = np.array([[1], [2], [3]])
      >>> points.shape, orders.shape
      ((3,), (3, 1))

      >>> modstruve(orders, points)
      array([[1.10275979e+00, 2.37282158e+01, 3.99247091e+02],
             [4.10260789e-01, 1.65535979e+01, 3.25973609e+02],
             [1.12472937e-01, 9.42430454e+00, 2.33544042e+02]])

  Plot the modified Struve functions of order 0 to 3 from -5 to 5.

      >>> fig, ax = plt.subplots()
      >>> x = np.linspace(-5., 5., 1000)
      >>> for i in range(4):
      ...     ax.plot(x, modstruve(i, x), label=f'$L_{i!r}$')
      >>> ax.legend(ncol=2)
      >>> ax.set_xlim(-5, 5)
      >>> ax.set_title(r"Modified Struve functions $L_{\nu}$")
      >>> plt.show()