scipy.special.struve#

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

Struve function.

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

\[H_v(x) = (z/2)^{v + 1} \sum_{n=0}^\infty \frac{(-1)^n (z/2)^{2n}}{\Gamma(n + \frac{3}{2}) \Gamma(n + v + \frac{3}{2})},\]

where \(\Gamma\) is the gamma function.

Parameters:

varray_like: Order of the 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:

Hscalar or ndarray: Value of the Struve function of order v at x.

See also

modstruve: Modified Struve function

Notes

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

power series
expansion in Bessel functions (if \(|z| < |v| + 20\))
asymptotic large-z expansion (if \(z \geq 0.7v + 12\))

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 Struve function of order 1 at 2.

>>> import numpy as np
>>> from scipy.special import struve
>>> import matplotlib.pyplot as plt
>>> struve(1, 2.)
0.6467637282835622

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

>>> struve([1, 2, 3], 2.)
array([0.64676373, 0.28031806, 0.08363767])

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

>>> points = np.array([2., 5., 8.])
>>> struve(1, points)
array([0.64676373, 0.80781195, 0.48811605])

Compute the 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))

>>> struve(orders, points)
array([[0.64676373, 0.80781195, 0.48811605],
       [0.28031806, 1.56937455, 1.51769363],
       [0.08363767, 1.50872065, 2.98697513]])

Plot the Struve functions of order 0 to 3 from -10 to 10.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> for i in range(4):
...     ax.plot(x, struve(i, x), label=f'$H_{i!r}$')
>>> ax.legend(ncol=2)
>>> ax.set_xlim(-10, 10)
>>> ax.set_title(r"Struve functions $H_{\nu}$")
>>> plt.show()

../../_images/scipy-special-struve-1.png