# 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:
varray_like

Order of the modified Struve function (float).

xarray_like

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

outndarray, optional

Optional output array for the function results

Returns:
Lscalar or ndarray

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

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()