Source code for sympy.functions.special.bessel

"""Bessel type functions"""

from __future__ import print_function, division

from sympy import S, pi, I
from sympy.core.function import Function, ArgumentIndexError, expand_func
from sympy.functions.elementary.trigonometric import sin, cos, csc, cot
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.complexes import re, im
from sympy.core.compatibility import xrange

# TODO
# o Airy Ai and Bi functions
# o Scorer functions G1 and G2
# o Asymptotic expansions
#   These are possible, e.g. for fixed order, but since the bessel type
#   functions are oscillatory they are not actually tractable at
#   infinity, so this is not particularly useful right now.
# o Series Expansions for functions of the second kind about zero
# o Nicer series expansions.
# o More rewriting.
# o Add solvers to ode.py (or rather add solvers for the hypergeometric equation).


[docs]class BesselBase(Function): """ Abstract base class for bessel-type functions. This class is meant to reduce code duplication. All Bessel type functions can 1) be differentiated, and the derivatives expressed in terms of similar functions and 2) be rewritten in terms of other bessel-type functions. Here "bessel-type functions" are assumed to have one complex parameter. To use this base class, define class attributes ``_a`` and ``_b`` such that ``2*F_n' = -_a*F_{n+1} + b*F_{n-1}``. """ nargs = 2 @property
[docs] def order(self): """ The order of the bessel-type function. """ return self.args[0]
@property
[docs] def argument(self): """ The argument of the bessel-type function. """ return self.args[1]
def fdiff(self, argindex=2): if argindex != 2: raise ArgumentIndexError(self, argindex) return self._b/2 * self.__class__(self.order - 1, self.argument) \ - self._a/2 * self.__class__(self.order + 1, self.argument) \ def _eval_conjugate(self): z = self.argument if (z.is_real and z.is_negative) is False: return self.__class__(self.order.conjugate(), z.conjugate()) def _eval_expand_func(self, **hints): nu, z, f = self.order, self.argument, self.__class__ if nu.is_real: if (nu - 1).is_positive: return (-self._a*self._b*f(nu - 2, z)._eval_expand_func() + 2*self._a*(nu - 1)*f(nu - 1, z)._eval_expand_func()/z) elif (nu + 1).is_negative: return (2*self._b*(nu + 1)*f(nu + 1, z)._eval_expand_func()/z - self._a*self._b*f(nu + 2, z)._eval_expand_func()) return self def _eval_simplify(self, ratio, measure): from sympy.simplify.simplify import besselsimp return besselsimp(self)
[docs]class besselj(BesselBase): r""" Bessel function of the first kind. The Bessel `J` function of order `\nu` is defined to be the function satisfying Bessel's differential equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0, with Laurent expansion .. math :: J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right), if :math:`\nu` is not a negative integer. If :math:`\nu=-n \in \mathbb{Z}_{<0}` *is* a negative integer, then the definition is .. math :: J_{-n}(z) = (-1)^n J_n(z). Examples ======== Create a Bessel function object: >>> from sympy import besselj, jn >>> from sympy.abc import z, n >>> b = besselj(n, z) Differentiate it: >>> b.diff(z) besselj(n - 1, z)/2 - besselj(n + 1, z)/2 Rewrite in terms of spherical Bessel functions: >>> b.rewrite(jn) sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi) Access the parameter and argument: >>> b.order n >>> b.argument z See Also ======== bessely, besseli, besselk References ========== .. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .. [2] Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 .. [3] http://en.wikipedia.org/wiki/Bessel_function .. [4] http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/ """ _a = S.One _b = S.One @classmethod def eval(cls, nu, z): if z.is_zero: if nu.is_zero: return S.One elif (nu.is_integer and nu.is_zero is False) or re(nu).is_positive: return S.Zero elif re(nu).is_negative and not (nu.is_integer is True): return S.ComplexInfinity elif nu.is_imaginary: return S.NaN if z is S.Infinity or (z is S.NegativeInfinity): return S.Zero if z.could_extract_minus_sign(): return (z)**nu*(-z)**(-nu)*besselj(nu, -z) if nu.is_integer: if nu.could_extract_minus_sign(): return S(-1)**(-nu)*besselj(-nu, z) newz = z.extract_multiplicatively(I) if newz: # NOTE we don't want to change the function if z==0 return I**(nu)*besseli(nu, newz) # branch handling: from sympy import unpolarify, exp if nu.is_integer: newz = unpolarify(z) if newz != z: return besselj(nu, newz) else: newz, n = z.extract_branch_factor() if n != 0: return exp(2*n*pi*nu*I)*besselj(nu, newz) nnu = unpolarify(nu) if nu != nnu: return besselj(nnu, z) def _eval_rewrite_as_besseli(self, nu, z): from sympy import polar_lift, exp return exp(I*pi*nu/2)*besseli(nu, polar_lift(-I)*z) def _eval_rewrite_as_bessely(self, nu, z): if nu.is_integer is False: return csc(pi*nu)*bessely(-nu, z) - cot(pi*nu)*bessely(nu, z) def _eval_rewrite_as_jn(self, nu, z): return sqrt(2*z/pi)*jn(nu - S.Half, self.argument) def _eval_is_real(self): nu, z = self.args if nu.is_integer and z.is_real: return True
[docs]class bessely(BesselBase): r""" Bessel function of the second kind. The Bessel `Y` function of order `\nu` is defined as .. math :: Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)}, where :math:`J_\mu(z)` is the Bessel function of the first kind. It is a solution to Bessel's equation, and linearly independent from :math:`J_\nu`. Examples ======== >>> from sympy import bessely, yn >>> from sympy.abc import z, n >>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi) See Also ======== besselj, besseli, besselk References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ """ _a = S.One _b = S.One @classmethod def eval(cls, nu, z): if z.is_zero: if nu.is_zero: return S.NegativeInfinity elif re(nu).is_zero is False: return S.ComplexInfinity elif re(nu).is_zero: return S.NaN if z is S.Infinity or z is S.NegativeInfinity: return S.Zero if nu.is_integer: if nu.could_extract_minus_sign(): return S(-1)**(-nu)*bessely(-nu, z) def _eval_rewrite_as_besselj(self, nu, z): if nu.is_integer is False: return csc(pi*nu)*(cos(pi*nu)*besselj(nu, z) - besselj(-nu, z)) def _eval_rewrite_as_besseli(self, nu, z): aj = self._eval_rewrite_as_besselj(*self.args) if aj: return aj.rewrite(besseli) def _eval_rewrite_as_yn(self, nu, z): return sqrt(2*z/pi) * yn(nu - S.Half, self.argument) def _eval_is_real(self): nu, z = self.args if nu.is_integer and z.is_positive: return True
[docs]class besseli(BesselBase): r""" Modified Bessel function of the first kind. The Bessel I function is a solution to the modified Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0. It can be defined as .. math :: I_\nu(z) = i^{-\nu} J_\nu(iz), where :math:`J_\nu(z)` is the Bessel function of the first kind. Examples ======== >>> from sympy import besseli >>> from sympy.abc import z, n >>> besseli(n, z).diff(z) besseli(n - 1, z)/2 + besseli(n + 1, z)/2 See Also ======== besselj, bessely, besselk References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ """ _a = -S.One _b = S.One @classmethod def eval(cls, nu, z): if z.is_zero: if nu.is_zero: return S.One elif (nu.is_integer and nu.is_zero is False) or re(nu).is_positive: return S.Zero elif re(nu).is_negative and not (nu.is_integer is True): return S.ComplexInfinity elif nu.is_imaginary: return S.NaN if z.is_imaginary: if im(z) is S.Infinity or im(z) is S.NegativeInfinity: return S.Zero if z.could_extract_minus_sign(): return (z)**nu*(-z)**(-nu)*besseli(nu, -z) if nu.is_integer: if nu.could_extract_minus_sign(): return besseli(-nu, z) newz = z.extract_multiplicatively(I) if newz: # NOTE we don't want to change the function if z==0 return I**(-nu)*besselj(nu, -newz) # branch handling: from sympy import unpolarify, exp if nu.is_integer: newz = unpolarify(z) if newz != z: return besseli(nu, newz) else: newz, n = z.extract_branch_factor() if n != 0: return exp(2*n*pi*nu*I)*besseli(nu, newz) nnu = unpolarify(nu) if nu != nnu: return besseli(nnu, z) def _eval_rewrite_as_besselj(self, nu, z): from sympy import polar_lift, exp return exp(-I*pi*nu/2)*besselj(nu, polar_lift(I)*z) def _eval_rewrite_as_bessely(self, nu, z): aj = self._eval_rewrite_as_besselj(*self.args) if aj: return aj.rewrite(bessely) def _eval_rewrite_as_jn(self, nu, z): return self._eval_rewrite_as_besselj(*self.args).rewrite(jn) def _eval_is_real(self): nu, z = self.args if nu.is_integer and z.is_real: return True
[docs]class besselk(BesselBase): r""" Modified Bessel function of the second kind. The Bessel K function of order :math:`\nu` is defined as .. math :: K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)}, where :math:`I_\mu(z)` is the modified Bessel function of the first kind. It is a solution of the modified Bessel equation, and linearly independent from :math:`Y_\nu`. Examples ======== >>> from sympy import besselk >>> from sympy.abc import z, n >>> besselk(n, z).diff(z) -besselk(n - 1, z)/2 - besselk(n + 1, z)/2 See Also ======== besselj, besseli, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ """ _a = S.One _b = -S.One @classmethod def eval(cls, nu, z): if z.is_zero: if nu.is_zero: return S.Infinity elif re(nu).is_zero is False: return S.ComplexInfinity elif re(nu).is_zero: return S.NaN if z.is_imaginary: if im(z) is S.Infinity or im(z) is S.NegativeInfinity: return S.Zero if nu.is_integer: if nu.could_extract_minus_sign(): return besselk(-nu, z) def _eval_rewrite_as_besseli(self, nu, z): if nu.is_integer is False: return pi*csc(pi*nu)*(besseli(-nu, z) - besseli(nu, z))/2 def _eval_rewrite_as_besselj(self, nu, z): ai = self._eval_rewrite_as_besseli(*self.args) if ai: return ai.rewrite(besselj) def _eval_rewrite_as_bessely(self, nu, z): aj = self._eval_rewrite_as_besselj(*self.args) if aj: return aj.rewrite(bessely) def _eval_rewrite_as_yn(self, nu, z): ay = self._eval_rewrite_as_bessely(*self.args) if ay: return ay.rewrite(yn) def _eval_is_real(self): nu, z = self.args if nu.is_integer and z.is_positive: return True
[docs]class hankel1(BesselBase): r""" Hankel function of the first kind. This function is defined as .. math :: H_\nu^{(1)} = J_\nu(z) + iY_\nu(z), where :math:`J_\nu(z)` is the Bessel function of the first kind, and :math:`Y_\nu(z)` is the Bessel function of the second kind. It is a solution to Bessel's equation. Examples ======== >>> from sympy import hankel1 >>> from sympy.abc import z, n >>> hankel1(n, z).diff(z) hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2 See Also ======== hankel2, besselj, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/ """ _a = S.One _b = S.One def _eval_conjugate(self): z = self.argument if (z.is_real and z.is_negative) is False: return hankel2(self.order.conjugate(), z.conjugate())
[docs]class hankel2(BesselBase): r""" Hankel function of the second kind. This function is defined as .. math :: H_\nu^{(2)} = J_\nu(z) - iY_\nu(z), where :math:`J_\nu(z)` is the Bessel function of the first kind, and :math:`Y_\nu(z)` is the Bessel function of the second kind. It is a solution to Bessel's equation, and linearly independent from :math:`H_\nu^{(1)}`. Examples ======== >>> from sympy import hankel2 >>> from sympy.abc import z, n >>> hankel2(n, z).diff(z) hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2 See Also ======== hankel1, besselj, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/ """ _a = S.One _b = S.One def _eval_conjugate(self): z = self.argument if (z.is_real and z.is_negative) is False: return hankel1(self.order.conjugate(), z.conjugate())
from sympy.polys.orthopolys import spherical_bessel_fn as fn class SphericalBesselBase(BesselBase): """ Base class for spherical Bessel functions. These are thin wrappers around ordinary Bessel functions, since spherical Bessel functions differ from the ordinary ones just by a slight change in order. To use this class, define the ``_rewrite`` and ``_expand`` methods. """ def _expand(self, **hints): """ Expand self into a polynomial. Nu is guaranteed to be Integer. """ raise NotImplementedError('expansion') def _rewrite(self): """ Rewrite self in terms of ordinary Bessel functions. """ raise NotImplementedError('rewriting') def _eval_expand_func(self, **hints): if self.order.is_Integer: return self._expand(**hints) else: return self def _eval_evalf(self, prec): return self._rewrite()._eval_evalf(prec) def fdiff(self, argindex=2): if argindex != 2: raise ArgumentIndexError(self, argindex) return self.__class__(self.order - 1, self.argument) - \ self * (self.order + 1)/self.argument
[docs]class jn(SphericalBesselBase): r""" Spherical Bessel function of the first kind. This function is a solution to the spherical Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0. It can be defined as .. math :: j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z), where :math:`J_\nu(z)` is the Bessel function of the first kind. Examples ======== >>> from sympy import Symbol, jn, sin, cos, expand_func >>> z = Symbol("z") >>> print(jn(0, z).expand(func=True)) sin(z)/z >>> jn(1, z).expand(func=True) == sin(z)/z**2 - cos(z)/z True >>> expand_func(jn(3, z)) (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z) The spherical Bessel functions of integral order are calculated using the formula: .. math:: j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z}, where the coefficients :math:`f_n(z)` are available as :func:`polys.orthopolys.spherical_bessel_fn`. See Also ======== besselj, bessely, besselk, yn """ def _rewrite(self): return self._eval_rewrite_as_besselj(self.order, self.argument) def _eval_rewrite_as_besselj(self, nu, z): return sqrt(pi/(2*z)) * besselj(nu + S('1/2'), z) def _expand(self, **hints): n = self.order z = self.argument return fn(n, z) * sin(z) + (-1)**(n + 1) * fn(-n - 1, z) * cos(z)
[docs]class yn(SphericalBesselBase): r""" Spherical Bessel function of the second kind. This function is another solution to the spherical Bessel equation, and linearly independent from :math:`j_n`. It can be defined as .. math :: j_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z), where :math:`Y_\nu(z)` is the Bessel function of the second kind. Examples ======== >>> from sympy import Symbol, yn, sin, cos, expand_func >>> z = Symbol("z") >>> print(expand_func(yn(0, z))) -cos(z)/z >>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z True For integral orders :math:`n`, :math:`y_n` is calculated using the formula: .. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z) See Also ======== besselj, bessely, besselk, jn """ def _rewrite(self): return self._eval_rewrite_as_bessely(self.order, self.argument) def _eval_rewrite_as_bessely(self, nu, z): return sqrt(pi/(2*z)) * bessely(nu + S('1/2'), z) def _expand(self, **hints): n = self.order z = self.argument return (-1)**(n + 1) * \ (fn(-n - 1, z) * sin(z) + (-1)**(-n) * fn(n, z) * cos(z))
[docs]def jn_zeros(n, k, method="sympy", dps=15): """ Zeros of the spherical Bessel function of the first kind. This returns an array of zeros of jn up to the k-th zero. * method = "sympy": uses :func:`mpmath.besseljzero` * method = "scipy": uses the `SciPy's sph_jn <http://docs.scipy.org/doc/scipy/reference/generated/scipy.special.jn.html>`_ and `newton <http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.newton.html>`_ to find all roots, which is faster than computing the zeros using a general numerical solver, but it requires SciPy and only works with low precision floating point numbers. [The function used with method="sympy" is a recent addition to mpmath, before that a general solver was used.] Examples ======== >>> from sympy import jn_zeros >>> jn_zeros(2, 4, dps=5) [5.7635, 9.095, 12.323, 15.515] See Also ======== jn, yn, besselj, besselk, bessely """ from math import pi if method == "sympy": from sympy.mpmath import besseljzero from sympy.mpmath.libmp.libmpf import dps_to_prec from sympy import Expr prec = dps_to_prec(dps) return [Expr._from_mpmath(besseljzero(S(n + 0.5)._to_mpmath(prec), int(k)), prec) for k in xrange(1, k + 1)] elif method == "scipy": from scipy.special import sph_jn from scipy.optimize import newton f = lambda x: sph_jn(n, x)[0][-1] else: raise NotImplementedError("Unknown method.") def solver(f, x): if method == "scipy": root = newton(f, x) else: raise NotImplementedError("Unknown method.") return root # we need to approximate the position of the first root: root = n + pi # determine the first root exactly: root = solver(f, root) roots = [root] for i in range(k - 1): # estimate the position of the next root using the last root + pi: root = solver(f, root + pi) roots.append(root) return roots