scipy.special.besselpoly

scipy.special.besselpoly(a, lmb, nu, out=None) = <wrapped_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.

Notes

Array API Standard Support

besselpoly has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. besselpoly does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[1]

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

Examples

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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()

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