Source code for mathematicskit.special_functions.systems.bessel
r"""Bessel functions of the first and second kind, via :mod:`scipy.special`.
``scipy.special.jv``/``yv`` implement these via well-tested series/
asymptotic expansions for any real order; mathematicskit does not reimplement
them. See Abramowitz & Stegun, *Handbook of Mathematical Functions*,
Ch. 9, and NIST *Digital Library of Mathematical Functions*, Ch. 10.
"""
from __future__ import annotations
from scipy import special
__all__ = ["bessel_first_kind", "bessel_second_kind"]
[docs]
def bessel_first_kind(nu, x):
r"""Bessel function of the first kind, :math:`J_\nu(x)`.
Solves Bessel's differential equation :math:`x^2y'' + xy' + (x^2 -
\nu^2)y = 0`, regular (finite) at :math:`x=0`. Via
:func:`scipy.special.jv`. See Abramowitz & Stegun, *Handbook of
Mathematical Functions*, Sec. 9.1.
Parameters
----------
nu : float
Order.
x : float or array-like of float
Returns
-------
float or ndarray
Examples
--------
>>> round(float(bessel_first_kind(0.0, 0.0)), 6) # J_0(0) = 1
1.0
>>> round(float(bessel_first_kind(1.0, 0.0)), 6) # J_nu(0) = 0 for nu > 0
0.0
"""
return special.jv(nu, x)
[docs]
def bessel_second_kind(nu, x):
r"""Bessel function of the second kind (Weber/Neumann function), :math:`Y_\nu(x)`.
The second, linearly independent solution of Bessel's equation,
singular (diverging to :math:`-\infty`) at :math:`x=0`. Via
:func:`scipy.special.yv`. See Abramowitz & Stegun, *Handbook of
Mathematical Functions*, Sec. 9.1.
Parameters
----------
nu : float
Order.
x : float or array-like of float
Returns
-------
float or ndarray
Examples
--------
>>> bool(bessel_second_kind(0.0, 0.1) < bessel_second_kind(0.0, 1.0)) # diverges toward x=0
True
"""
return special.yv(nu, x)
