# scipy.special.besselpoly#

scipy.special.besselpoly(a, lmb, nu, out=None) = <ufunc 'besselpoly'>#

Weighted integral of the Bessel function of the first kind.

Computes

$\int_0^1 x^\lambda J_\nu(2 a x) \, dx$

where $$J_\nu$$ is a Bessel function and $$\lambda=lmb$$, $$\nu=nu$$.

Parameters:
aarray_like

Scale factor inside the Bessel function.

lmbarray_like

Power of x

nuarray_like

Order of the Bessel function.

outndarray, optional

Optional output array for the function results.

Returns:
scalar or ndarray

Value of the integral.

References

[1]

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

Examples

Evaluate the function for one parameter set.

>>> from scipy.special import besselpoly
>>> besselpoly(1, 1, 1)
0.24449718372863877


Evaluate the function for different scale factors.

>>> import numpy as np
>>> factors = np.array([0., 3., 6.])
>>> besselpoly(factors, 1, 1)
array([ 0.        , -0.00549029,  0.00140174])


Plot the function for varying powers, orders and scales.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> powers = np.linspace(0, 10, 100)
>>> orders = [1, 2, 3]
>>> scales = [1, 2]
>>> all_combinations = [(order, scale) for order in orders
...                     for scale in scales]
>>> for order, scale in all_combinations:
...     ax.plot(powers, besselpoly(scale, powers, order),
...             label=rf"$\nu={order}, a={scale}$")
>>> ax.legend()
>>> ax.set_xlabel(r"$\lambda$")
>>> ax.set_ylabel(r"$\int_0^1 x^{\lambda} J_{\nu}(2ax)\,dx$")
>>> plt.show()