# Source code for diofant.functions.special.bessel

from mpmath import besseljzero, mp, workprec
from mpmath.libmp.libmpf import dps_to_prec

from ...core import (Add, Expr, Function, I, Integer, Pow, Rational, Wild,
cacheit, nan, oo, pi, sympify, zoo)
from ...core.function import ArgumentIndexError
from ...polys.orthopolys import spherical_bessel_fn as fn
from ..combinatorial.factorials import factorial
from ..elementary.complexes import Abs, im, re
from ..elementary.exponential import exp
from ..elementary.miscellaneous import cbrt, root, sqrt
from ..elementary.trigonometric import cos, cot, csc, sin
from .gamma_functions import gamma
from .hyper import hyper

# TODO
# 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}.

"""

@property
def order(self):
"""The order of the bessel-type function."""
return self.args[0]

@property
def argument(self):
"""The argument of the bessel-type function."""
return self.args[1]

@classmethod
def eval(cls, nu, z):
return

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_extended_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_extended_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 ...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 \nu is not a negative integer. If \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:

>>> 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

========

bessely, besseli, besselk

References
==========

* Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9",
Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables
* Luke, Y. L. (1969), The Special Functions and Their
Approximations, Volume 1
* https://en.wikipedia.org/wiki/Bessel_function
* http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/

"""

_a = Integer(1)
_b = Integer(1)

@classmethod
def eval(cls, nu, z):
if z.is_zero:
if nu.is_zero:
return Integer(1)
elif (nu.is_integer and nu.is_nonzero) or re(nu).is_positive:
return Integer(0)
elif re(nu).is_negative and not (nu.is_integer is True):
return zoo
elif nu.is_imaginary:
return nan
if z in (oo, -oo):
return Integer(0)

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 Integer(-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 .. import unpolarify
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 .. import polar_lift
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 - Rational(1, 2), self.argument)

def _eval_is_extended_real(self):
nu, z = self.args
if nu.is_integer and z.is_extended_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 J_\mu(z) is the Bessel function of the first kind.

It is a solution to Bessel's equation, and linearly independent from
J_\nu.

Examples
========

>>> 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)

========

besselj, besseli, besselk

References
==========

* http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/

"""

_a = Integer(1)
_b = Integer(1)

@classmethod
def eval(cls, nu, z):
if z.is_zero:
if nu.is_zero:
return -oo
elif re(nu).is_nonzero:
return zoo
elif re(nu).is_zero:
return nan
if z in (oo, -oo):
return Integer(0)

if nu.is_integer:
if nu.could_extract_minus_sign():
return Integer(-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 - Rational(1, 2), self.argument)

def _eval_is_extended_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 J_\nu(z) is the Bessel function of the first kind.

Examples
========

>>> besseli(n, z).diff(z)
besseli(n - 1, z)/2 + besseli(n + 1, z)/2

========

besselj, bessely, besselk

References
==========

* http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/

"""

_a = -Integer(1)
_b = Integer(1)

@classmethod
def eval(cls, nu, z):
if z.is_zero:
if nu.is_zero:
return Integer(1)
elif (nu.is_integer and nu.is_nonzero) or re(nu).is_positive:
return Integer(0)
elif re(nu).is_negative and not (nu.is_integer is True):
return zoo
elif nu.is_imaginary:
return nan
if im(z) in (oo, -oo):
return Integer(0)

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 .. import unpolarify
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 .. import polar_lift
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)
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_extended_real(self):
nu, z = self.args
if nu.is_integer and z.is_extended_real:
return True

[docs]class besselk(BesselBase):
r"""
Modified Bessel function of the second kind.

The Bessel K function of order \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 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 Y_\nu.

Examples
========

>>> besselk(n, z).diff(z)
-besselk(n - 1, z)/2 - besselk(n + 1, z)/2

========

besselj, besseli, bessely

References
==========

* http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/

"""

_a = Integer(1)
_b = -Integer(1)

@classmethod
def eval(cls, nu, z):
if z.is_zero:
if nu.is_zero:
return oo
elif re(nu).is_nonzero:
return zoo
elif re(nu).is_zero:
return nan
if im(z) in (oo, -oo):
return Integer(0)

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_is_extended_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 J_\nu(z) is the Bessel function of the first kind, and
Y_\nu(z) is the Bessel function of the second kind.

It is a solution to Bessel's equation.

Examples
========

>>> hankel1(n, z).diff(z)
hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2

========

hankel2, besselj, bessely

References
==========

* http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/

"""

_a = Integer(1)
_b = Integer(1)

def _eval_conjugate(self):
z = self.argument
if (z.is_extended_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 J_\nu(z) is the Bessel function of the first kind, and
Y_\nu(z) is the Bessel function of the second kind.

It is a solution to Bessel's equation, and linearly independent from
H_\nu^{(1)}.

Examples
========

>>> hankel2(n, z).diff(z)
hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2

========

hankel1, besselj, bessely

References
==========

* http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/

"""

_a = Integer(1)
_b = Integer(1)

def _eval_conjugate(self):
z = self.argument
if (z.is_extended_real and z.is_negative) is False:
return hankel1(self.order.conjugate(), z.conjugate())

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 J_\nu(z) is the Bessel function of the first kind.

Examples
========

>>> 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 f_n(z) are available as
:func:diofant.polys.orthopolys.spherical_bessel_fn.

========

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 + Rational(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 j_n. It can be defined as

.. math ::
j_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z),

where Y_\nu(z) is the Bessel function of the second kind.

Examples
========

>>> expand_func(yn(0, z))
-cos(z)/z
>>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z
True

For integral orders n, y_n is calculated using the formula:

.. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z)

========

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 + Rational(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="diofant", 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 = "diofant": uses mpmath's function besseljzero
* method = "scipy": uses :func:scipy.special.jn_zeros.
and :func:scipy.optimize.newton 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="diofant" is a recent addition to mpmath, before that a general
solver was used.]

Examples
========

>>> jn_zeros(2, 4, dps=5)
[5.7635, 9.095, 12.323, 15.515]

========

jn, yn, besselj, besselk, bessely

"""
from math import pi

if method == "diofant":
prec = dps_to_prec(dps)
return [Expr._from_mpmath(besseljzero(sympify(n + 0.5)._to_mpmath(prec),
int(l)), prec)
for l in range(1, k + 1)]
elif method == "scipy":
from scipy.optimize import newton
try:
from scipy.special import spherical_jn
except ImportError:  # pragma: no cover
from scipy.special import sph_jn

def spherical_jn(n, x):
return sph_jn(n, x)[0][-1]

def f(x):
return spherical_jn(n, x)
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

[docs]class AiryBase(Function):
"""
Abstract base class for Airy functions.

This class is meant to reduce code duplication.

"""

def _eval_conjugate(self):
return self.func(self.args[0].conjugate())

def _eval_is_extended_real(self):
if self.args[0].is_extended_real:
return True

def as_real_imag(self, deep=True, **hints):
if self.args[0].is_extended_real:
if deep:
hints['complex'] = False
return self.expand(deep, **hints), Integer(0)
else:
return self, Integer(0)
if deep:
x, y = self.args[0].expand(deep, **hints).as_real_imag()
else:
x, y = self.args[0].as_real_imag()

sq = -y**2/x**2
re = (self.func(x+x*sqrt(sq)) + self.func(x-x*sqrt(sq)))/2
im = x/(2*y) * sqrt(sq) * (self.func(x-x*sqrt(sq)) - self.func(x+x*sqrt(sq)))
return re, im

[docs]class airyai(AiryBase):
r"""
The Airy function \operatorname{Ai} of the first kind.

The Airy function \operatorname{Ai}(z) is defined to be the function
satisfying Airy's differential equation

.. math::
\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.

Equivalently, for real z

.. math::
\operatorname{Ai}(z) := \frac{1}{\pi}
\int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.

Examples
========

Create an Airy function object:

>>> airyai(z)
airyai(z)

Several special values are known:

>>> airyai(0)
3**(1/3)/(3*gamma(2/3))
>>> airyai(oo)
0
>>> airyai(-oo)
0

The Airy function obeys the mirror symmetry:

>>> conjugate(airyai(z))
airyai(conjugate(z))

Differentiation with respect to z is supported:

>>> diff(airyai(z), z)
airyaiprime(z)
>>> diff(airyai(z), z, 2)
z*airyai(z)

Series expansion is also supported:

>>> series(airyai(z), z, 0, 3)
3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision
on the whole complex plane:

>>> airyai(-2).evalf(50)
0.22740742820168557599192443603787379946077222541710

Rewrite Ai(z) in terms of hypergeometric functions:

>>> airyai(z).rewrite(hyper)
-3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))

========

airybi: Airy function of the second kind.
airyaiprime: Derivative of the Airy function of the first kind.
airybiprime: Derivative of the Airy function of the second kind.

References
==========

* https://en.wikipedia.org/wiki/Airy_function
* https://dlmf.nist.gov/9
* https://www.encyclopediaofmath.org/index.php/Airy_functions
* http://mathworld.wolfram.com/AiryFunctions.html

"""

unbranched = True

@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg in (oo, -oo):
return Integer(0)
elif arg == 0:
return 1/(3**Rational(2, 3) * gamma(Rational(2, 3)))

def fdiff(self, argindex=1):
if argindex == 1:
return airyaiprime(self.args[0])
else:
raise ArgumentIndexError(self, argindex)

@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0:
return Integer(0)
else:
x = sympify(x)
return 1/(3**Rational(2, 3)*pi) * gamma(Rational(n+1, 3)) * sin(2*pi*(n+1)/3) / factorial(n) * (root(3, 3)*x)**n

def _eval_rewrite_as_besselj(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(-z, Rational(3, 2))
if re(z).is_negative:
return ot*sqrt(-z) * (besselj(-ot, tt*a) + besselj(ot, tt*a))

def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(z, Rational(3, 2))
if re(z).is_positive:
return ot*sqrt(z) * (besseli(-ot, tt*a) - besseli(ot, tt*a))
else:
return ot*(Pow(a, ot)*besseli(-ot, tt*a) - z*Pow(a, -ot)*besseli(ot, tt*a))

def _eval_rewrite_as_hyper(self, z):
pf1 = 1/(3**Rational(2, 3)*gamma(Rational(2, 3)))
pf2 = z / (root(3, 3)*gamma(Rational(1, 3)))
return pf1 * hyper([], [Rational(2, 3)], z**3/9) - pf2 * hyper([], [Rational(4, 3)], z**3/9)

def _eval_rewrite_as_tractable(self, z):
return exp(-Rational(2, 3)*z**Rational(3, 2))*sqrt(pi*sqrt(z))/2*_airyais(z)

def _eval_expand_func(self, **hints):
arg = self.args[0]
symbs = arg.free_symbols

if len(symbs) == 1:
z = symbs.pop()
c = Wild("c", exclude=[z])
d = Wild("d", exclude=[z])
m = Wild("m", exclude=[z])
n = Wild("n", exclude=[z])
M = arg.match(c*(d*z**n)**m)
if M is not None:
m = M[m]
# The transformation is given by 03.05.16.0001.01
# http://functions.wolfram.com/Bessel-TypeFunctions/AiryAi/16/01/01/0001/
if (3*m).is_integer:
c = M[c]
d = M[d]
n = M[n]
pf = (d * z**n)**m / (d**m * z**(m*n))
newarg = c * d**m * z**(m*n)
return ((pf + 1)*airyai(newarg) - (pf - 1)/sqrt(3)*airybi(newarg))/2
return self.func(*self.args)

[docs]class airybi(AiryBase):
r"""
The Airy function \operatorname{Bi} of the second kind.

The Airy function \operatorname{Bi}(z) is defined to be the function
satisfying Airy's differential equation

.. math::
\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.

Equivalently, for real z

.. math::
\operatorname{Bi}(z) := \frac{1}{\pi}
\int_0^\infty
\exp\left(-\frac{t^3}{3} + z t\right)
+ \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.

Examples
========

Create an Airy function object:

>>> airybi(z)
airybi(z)

Several special values are known:

>>> airybi(0)
3**(5/6)/(3*gamma(2/3))
>>> airybi(oo)
oo
>>> airybi(-oo)
0

The Airy function obeys the mirror symmetry:

>>> conjugate(airybi(z))
airybi(conjugate(z))

Differentiation with respect to z is supported:

>>> diff(airybi(z), z)
airybiprime(z)
>>> diff(airybi(z), z, 2)
z*airybi(z)

Series expansion is also supported:

>>> series(airybi(z), z, 0, 3)
3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision
on the whole complex plane:

>>> airybi(-2).evalf(50)
-0.41230258795639848808323405461146104203453483447240

Rewrite Bi(z) in terms of hypergeometric functions:

>>> airybi(z).rewrite(hyper)
3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))

========

airyai: Airy function of the first kind.
airyaiprime: Derivative of the Airy function of the first kind.
airybiprime: Derivative of the Airy function of the second kind.

References
==========

* https://en.wikipedia.org/wiki/Airy_function
* https://dlmf.nist.gov/9
* https://www.encyclopediaofmath.org/index.php/Airy_functions
* http://mathworld.wolfram.com/AiryFunctions.html

"""

unbranched = True

@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is oo:
return oo
elif arg == -oo:
return Integer(0)
elif arg == 0:
return 1/(root(3, 6)*gamma(Rational(2, 3)))

def fdiff(self, argindex=1):
if argindex == 1:
return airybiprime(self.args[0])
else:
raise ArgumentIndexError(self, argindex)

@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0:
return Integer(0)
else:
x = sympify(x)
return 1/(root(3, 6)*pi) * gamma(Rational(n + 1, 3)) * Abs(sin(2*pi*(n + 1)/3)) / factorial(n) * (root(3, 3)*x)**n

def _eval_rewrite_as_besselj(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(-z, Rational(3, 2))
if re(z).is_negative:
return sqrt(-z/3) * (besselj(-ot, tt*a) - besselj(ot, tt*a))

def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = Pow(z, Rational(3, 2))
if re(z).is_positive:
return sqrt(z)/sqrt(3) * (besseli(-ot, tt*a) + besseli(ot, tt*a))
else:
b = Pow(a, ot)
c = Pow(a, -ot)
return sqrt(ot)*(b*besseli(-ot, tt*a) + z*c*besseli(ot, tt*a))

def _eval_rewrite_as_hyper(self, z):
pf1 = 1/(root(3, 6)*gamma(Rational(2, 3)))
pf2 = z*root(3, 6) / gamma(Rational(1, 3))
return pf1 * hyper([], [Rational(2, 3)], z**3/9) + pf2 * hyper([], [Rational(4, 3)], z**3/9)

def _eval_rewrite_as_tractable(self, z):
return exp(Rational(2, 3)*z**Rational(3, 2))*sqrt(pi*sqrt(z))*_airybis(z)

def _eval_expand_func(self, **hints):
arg = self.args[0]
symbs = arg.free_symbols

if len(symbs) == 1:
z = symbs.pop()
c = Wild("c", exclude=[z])
d = Wild("d", exclude=[z])
m = Wild("m", exclude=[z])
n = Wild("n", exclude=[z])
M = arg.match(c*(d*z**n)**m)
if M is not None:
m = M[m]
# The transformation is given by 03.06.16.0001.01
# http://functions.wolfram.com/Bessel-TypeFunctions/AiryBi/16/01/01/0001/
if (3*m).is_integer:
c = M[c]
d = M[d]
n = M[n]
pf = (d * z**n)**m / (d**m * z**(m*n))
newarg = c * d**m * z**(m*n)
return (sqrt(3)*(1 - pf)*airyai(newarg) + (1 + pf)*airybi(newarg))/2
return self.func(*self.args)

class _airyais(Function):
def _eval_rewrite_as_intractable(self, x):
return 2*airyai(x)*exp(Rational(2, 3)*x**Rational(3, 2))/sqrt(pi*sqrt(x))

def _eval_aseries(self, n, args0, x, logx):
from ...simplify import combsimp
from ...series import Order
point = args0[0]

if point is oo:
z = self.args[0]
l = [gamma(k + Rational(5, 6))*gamma(k + Rational(1, 6)) *
Rational(-3, 4)**k/(2*pi**2*factorial(k) *
z**Rational(3*k + 1, 2)) for k in range(n)]
l = [combsimp(t) for t in l]
o = Order(1/z**Rational(3*n + 1, 2), x)
return Add(*l)._eval_nseries(x, n, logx) + o

# All other points are not handled
return super()._eval_aseries(n, args0, x, logx)

def _eval_nseries(self, x, n, logx):
x0 = self.args[0].limit(x, 0)
if x0 == 0:
return self.rewrite('intractable')._eval_nseries(x, n, logx)
else:
return super()._eval_nseries(x, n, logx)

class _airybis(Function):
def _eval_rewrite_as_intractable(self, x):
return airybi(x)*exp(-Rational(2, 3)*x**Rational(3, 2))/sqrt(pi*sqrt(x))

def _eval_aseries(self, n, args0, x, logx):
from ...simplify import combsimp
from ...series import Order
point = args0[0]

if point is oo:
z = self.args[0]
l = [gamma(k + Rational(5, 6))*gamma(k + Rational(1, 6)) *
Rational(3, 4)**k/(2*pi**2*factorial(k) *
z**Rational(3*k + 1, 2)) for k in range(n)]
l = [combsimp(t) for t in l]
o = Order(1/z**Rational(3*n + 1, 2), x)
return Add(*l)._eval_nseries(x, n, logx) + o

# All other points are not handled
return super()._eval_aseries(n, args0, x, logx)

def _eval_nseries(self, x, n, logx):
x0 = self.args[0].limit(x, 0)
if x0 == 0:
return self.rewrite('intractable')._eval_nseries(x, n, logx)
else:
return super()._eval_nseries(x, n, logx)

[docs]class airyaiprime(AiryBase):
r"""
The derivative \operatorname{Ai}^\prime of the Airy function of the first kind.

The Airy function \operatorname{Ai}^\prime(z) is defined to be the function

.. math::
\operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}.

Examples
========

Create an Airy function object:

>>> airyaiprime(z)
airyaiprime(z)

Several special values are known:

>>> airyaiprime(0)
-3**(2/3)/(3*gamma(1/3))
>>> airyaiprime(oo)
0

The Airy function obeys the mirror symmetry:

>>> conjugate(airyaiprime(z))
airyaiprime(conjugate(z))

Differentiation with respect to z is supported:

>>> diff(airyaiprime(z), z)
z*airyai(z)
>>> diff(airyaiprime(z), z, 2)
z*airyaiprime(z) + airyai(z)

Series expansion is also supported:

>>> series(airyaiprime(z), z, 0, 3)
-3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision
on the whole complex plane:

>>> airyaiprime(-2).evalf(50)
0.61825902074169104140626429133247528291577794512415

Rewrite Ai'(z) in terms of hypergeometric functions:

>>> airyaiprime(z).rewrite(hyper)
3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3))

========

airyai: Airy function of the first kind.
airybi: Airy function of the second kind.
airybiprime: Derivative of the Airy function of the second kind.

References
==========

* https://en.wikipedia.org/wiki/Airy_function
* https://dlmf.nist.gov/9
* https://www.encyclopediaofmath.org/index.php/Airy_functions
* http://mathworld.wolfram.com/AiryFunctions.html

"""

unbranched = True

@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is oo:
return Integer(0)
elif arg == 0:
return -1/(cbrt(3) * gamma(Rational(1, 3)))

def fdiff(self, argindex=1):
if argindex == 1:
return self.args[0]*airyai(self.args[0])
else:
raise ArgumentIndexError(self, argindex)

def _eval_evalf(self, prec):
z = self.args[0]._to_mpmath(prec)
with workprec(prec):
res = mp.airyai(z, derivative=1)
return Expr._from_mpmath(res, prec)

def _eval_rewrite_as_besselj(self, z):
tt = Rational(2, 3)
a = Pow(-z, Rational(3, 2))
if re(z).is_negative:
return z/3 * (besselj(-tt, tt*a) - besselj(tt, tt*a))

def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = tt * Pow(z, Rational(3, 2))
if re(z).is_positive:
return z/3 * (besseli(tt, a) - besseli(-tt, a))
else:
a = Pow(z, Rational(3, 2))
b = Pow(a, tt)
c = Pow(a, -tt)
return ot * (z**2*c*besseli(tt, tt*a) - b*besseli(-ot, tt*a))

def _eval_rewrite_as_hyper(self, z):
pf1 = z**2 / (2*3**Rational(2, 3)*gamma(Rational(2, 3)))
pf2 = 1 / (root(3, 3)*gamma(Rational(1, 3)))
return pf1 * hyper([], [Rational(5, 3)], z**3/9) - pf2 * hyper([], [Rational(1, 3)], z**3/9)

def _eval_expand_func(self, **hints):
arg = self.args[0]
symbs = arg.free_symbols

if len(symbs) == 1:
z = symbs.pop()
c = Wild("c", exclude=[z])
d = Wild("d", exclude=[z])
m = Wild("m", exclude=[z])
n = Wild("n", exclude=[z])
M = arg.match(c*(d*z**n)**m)
if M is not None:
m = M[m]
# The transformation is in principle
# given by 03.07.16.0001.01 but note
# that there is an error in this formule.
# http://functions.wolfram.com/Bessel-TypeFunctions/AiryAiPrime/16/01/01/0001/
if (3*m).is_integer:
c = M[c]
d = M[d]
n = M[n]
pf = (d**m * z**(n*m)) / (d * z**n)**m
newarg = c * d**m * z**(n*m)
return ((pf + 1)*airyaiprime(newarg) + (pf - 1)/sqrt(3)*airybiprime(newarg))/2
return self.func(*self.args)

[docs]class airybiprime(AiryBase):
r"""
The derivative \operatorname{Bi}^\prime of the Airy function of the first kind.

The Airy function \operatorname{Bi}^\prime(z) is defined to be the function

.. math::
\operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}.

Examples
========

Create an Airy function object:

>>> airybiprime(z)
airybiprime(z)

Several special values are known:

>>> airybiprime(0)
3**(1/6)/gamma(1/3)
>>> airybiprime(oo)
oo
>>> airybiprime(-oo)
0

The Airy function obeys the mirror symmetry:

>>> conjugate(airybiprime(z))
airybiprime(conjugate(z))

Differentiation with respect to z is supported:

>>> diff(airybiprime(z), z)
z*airybi(z)
>>> diff(airybiprime(z), z, 2)
z*airybiprime(z) + airybi(z)

Series expansion is also supported:

>>> series(airybiprime(z), z, 0, 3)
3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3)

We can numerically evaluate the Airy function to arbitrary precision
on the whole complex plane:

>>> airybiprime(-2).evalf(50)
0.27879516692116952268509756941098324140300059345163

Rewrite Bi'(z) in terms of hypergeometric functions:

>>> airybiprime(z).rewrite(hyper)
3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3)

========

airyai: Airy function of the first kind.
airybi: Airy function of the second kind.
airyaiprime: Derivative of the Airy function of the first kind.

References
==========

* https://en.wikipedia.org/wiki/Airy_function
* https://dlmf.nist.gov/9
* https://www.encyclopediaofmath.org/index.php/Airy_functions
* http://mathworld.wolfram.com/AiryFunctions.html

"""

unbranched = True

@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is oo:
return oo
elif arg == -oo:
return Integer(0)
elif arg == 0:
return root(3, 6)/gamma(Rational(1, 3))

def fdiff(self, argindex=1):
if argindex == 1:
return self.args[0]*airybi(self.args[0])
else:
raise ArgumentIndexError(self, argindex)

def _eval_evalf(self, prec):
z = self.args[0]._to_mpmath(prec)
with workprec(prec):
res = mp.airybi(z, derivative=1)
return Expr._from_mpmath(res, prec)

def _eval_rewrite_as_besselj(self, z):
tt = Rational(2, 3)
a = tt * Pow(-z, Rational(3, 2))
if re(z).is_negative:
return -z/sqrt(3) * (besselj(-tt, a) + besselj(tt, a))

def _eval_rewrite_as_besseli(self, z):
ot = Rational(1, 3)
tt = Rational(2, 3)
a = tt * Pow(z, Rational(3, 2))
if re(z).is_positive:
return z/sqrt(3) * (besseli(-tt, a) + besseli(tt, a))
else:
a = Pow(z, Rational(3, 2))
b = Pow(a, tt)
c = Pow(a, -tt)
return sqrt(ot) * (b*besseli(-tt, tt*a) + z**2*c*besseli(tt, tt*a))

def _eval_rewrite_as_hyper(self, z):
pf1 = z**2 / (2*root(3, 6)*gamma(Rational(2, 3)))
pf2 = root(3, 6) / gamma(Rational(1, 3))
return pf1 * hyper([], [Rational(5, 3)], z**3/9) + pf2 * hyper([], [Rational(1, 3)], z**3/9)

def _eval_expand_func(self, **hints):
arg = self.args[0]
symbs = arg.free_symbols

if len(symbs) == 1:
z = symbs.pop()
c = Wild("c", exclude=[z])
d = Wild("d", exclude=[z])
m = Wild("m", exclude=[z])
n = Wild("n", exclude=[z])
M = arg.match(c*(d*z**n)**m)
if M is not None:
m = M[m]
# The transformation is in principle
# given by 03.08.16.0001.01 but note
# that there is an error in this formule.
# http://functions.wolfram.com/Bessel-TypeFunctions/AiryBiPrime/16/01/01/0001/
if (3*m).is_integer:
c = M[c]
d = M[d]
n = M[n]
pf = (d**m * z**(n*m)) / (d * z**n)**m
newarg = c * d**m * z**(n*m)
return (sqrt(3)*(pf - 1)*airyaiprime(newarg) + (pf + 1)*airybiprime(newarg))/2
return self.func(*self.args)