r"""
Special Functions
AUTHORS:
- David Joyner (2006-13-06): initial version
- David Joyner (2006-30-10): bug fixes to pari wrappers of Bessel
functions, hypergeometric_U
- William Stein (2008-02): Impose some sanity checks.
- David Joyner (2008-04-23): addition of elliptic integrals
This module provides easy access to many of Maxima and PARI's
special functions.
Maxima's special functions package (which includes spherical
harmonic functions, spherical Bessel functions (of the 1st and 2nd
kind), and spherical Hankel functions (of the 1st and 2nd kind))
was written by Barton Willis of the University of Nebraska at
Kearney. It is released under the terms of the General Public
License (GPL).
Support for elliptic functions and integrals was written by Raymond
Toy. It is placed under the terms of the General Public License
(GPL) that governs the distribution of Maxima.
The (usual) Bessel functions and Airy functions are part of the
standard Maxima package. Some Bessel functions also are implemented
in PARI. (Caution: The PARI versions are sometimes different than
the Maxima version.) For example, the J-Bessel function
`J_\nu (z)` can be computed using either Maxima or PARI,
depending on an optional variable you pass to bessel_J.
Next, we summarize some of the properties of the functions
implemented here.
- Bessel functions, first defined by the Swiss mathematician
Daniel Bernoulli and named after Friedrich Bessel, are canonical
solutions y(x) of Bessel's differential equation:
.. math::
x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0,
for an arbitrary real number `\alpha` (the order).
- Another important formulation of the two linearly independent
solutions to Bessel's equation are the Hankel functions
`H_\alpha^{(1)}(x)` and `H_\alpha^{(2)}(x)`,
defined by:
.. math::
H_\alpha^{(1)}(x) = J_\alpha(x) + i Y_\alpha(x)
.. math::
H_\alpha^{(2)}(x) = J_\alpha(x) - i Y_\alpha(x)
where `i` is the imaginary unit (and `J_*` and
`Y_*` are the usual J- and Y-Bessel functions). These
linear combinations are also known as Bessel functions of the third
kind; they are two linearly independent solutions of Bessel's
differential equation. They are named for Hermann Hankel.
- Airy function The function `Ai(x)` and the related
function `Bi(x)`, which is also called an Airy function,
are solutions to the differential equation
.. math::
y'' - xy = 0,
known as the Airy equation. They belong to the class of 'Bessel functions of
fractional order'. The initial conditions
`Ai(0) = (\Gamma(2/3)3^{2/3})^{-1}`,
`Ai'(0) = -(\Gamma(1/3)3^{1/3})^{-1}` define
`Ai(x)`. The initial conditions
`Bi(0) = 3^{1/2}Ai(0)`, `Bi'(0) = -3^{1/2}Ai'(0)`
define `Bi(x)`.
They are named after the British astronomer George Biddell Airy.
- Spherical harmonics: Laplace's equation in spherical coordinates
is:
.. math::
{\frac{1}{r^2}}{\frac{\partial}{\partial r}} \left(r^2 {\frac{\partial f}{\partial r}}\right) + {\frac{1}{r^2}\sin\theta}{\frac{\partial}{\partial \theta}} \left(\sin\theta {\frac{\partial f}{\partial \theta}}\right) + {\frac{1}{r^2\sin^2\theta}}{\frac{\partial^2 f}{\partial \varphi^2}} = 0.
Note that the spherical coordinates `\theta` and
`\varphi` are defined here as follows: `\theta` is
the colatitude or polar angle, ranging from
`0\leq\theta\leq\pi` and `\varphi` the azimuth or
longitude, ranging from `0\leq\varphi<2\pi`.
The general solution which remains finite towards infinity is a
linear combination of functions of the form
.. math::
r^{-1-\ell} \cos (m \varphi) P_\ell^m (\cos{\theta} )
and
.. math::
r^{-1-\ell} \sin (m \varphi) P_\ell^m (\cos{\theta} )
where `P_\ell^m` are the associated Legendre polynomials,
and with integer parameters `\ell \ge 0` and `m`
from `0` to `\ell`. Put in another way, the
solutions with integer parameters `\ell \ge 0` and
`- \ell\leq m\leq \ell`, can be written as linear
combinations of:
.. math::
U_{\ell,m}(r,\theta , \varphi ) = r^{-1-\ell} Y_\ell^m( \theta , \varphi )
where the functions `Y` are the spherical harmonic
functions with parameters `\ell`, `m`, which can be
written as:
.. math::
Y_\ell^m( \theta , \varphi ) = \sqrt{{\frac{(2\ell+1)}{4\pi}}{\frac{(\ell-m)!}{(\ell+m)!}}} \cdot e^{i m \varphi } \cdot P_\ell^m ( \cos{\theta} ) .
The spherical harmonics obey the normalisation condition
.. math::
\int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi} Y_\ell^mY_{\ell'}^{m'*}\,d\Omega =\delta_{\ell\ell'}\delta_{mm'}\quad\quad d\Omega =\sin\theta\,d\varphi\,d\theta .
- When solving for separable solutions of Laplace's equation in
spherical coordinates, the radial equation has the form:
.. math::
x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} + [x^2 - n(n+1)]y = 0.
The spherical Bessel functions `j_n` and `y_n`,
are two linearly independent solutions to this equation. They are
related to the ordinary Bessel functions `J_n` and
`Y_n` by:
.. math::
j_n(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x),
.. math::
y_n(x) = \sqrt{\frac{\pi}{2x}} Y_{n+1/2}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-1/2}(x).
- For `x>0`, the confluent hypergeometric function
`y = U(a,b,x)` is defined to be the solution to Kummer's
differential equation
.. math::
xy'' + (b-x)y' - ay = 0,
which satisfies `U(a,b,x) \sim x^{-a}`, as
`x\rightarrow \infty`. (There is a linearly independent
solution, called Kummer's function `M(a,b,x)`, which is not
implemented.)
- Jacobi elliptic functions can be thought of as generalizations
of both ordinary and hyperbolic trig functions. There are twelve
Jacobian elliptic functions. Each of the twelve corresponds to an
arrow drawn from one corner of a rectangle to another.
::
n ------------------- d
| |
| |
| |
s ------------------- c
Each of the corners of the rectangle are labeled, by convention, s,
c, d and n. The rectangle is understood to be lying on the complex
plane, so that s is at the origin, c is on the real axis, and n is
on the imaginary axis. The twelve Jacobian elliptic functions are
then pq(x), where p and q are one of the letters s,c,d,n.
The Jacobian elliptic functions are then the unique
doubly-periodic, meromorphic functions satisfying the following
three properties:
#. There is a simple zero at the corner p, and a simple pole at the
corner q.
#. The step from p to q is equal to half the period of the function
pq(x); that is, the function pq(x) is periodic in the direction pq,
with the period being twice the distance from p to q. Also, pq(x)
is also periodic in the other two directions as well, with a period
such that the distance from p to one of the other corners is a
quarter period.
#. If the function pq(x) is expanded in terms of x at one of the
corners, the leading term in the expansion has a coefficient of 1.
In other words, the leading term of the expansion of pq(x) at the
corner p is x; the leading term of the expansion at the corner q is
1/x, and the leading term of an expansion at the other two corners
is 1.
We can write
.. math::
pq(x)=\frac{pr(x)}{qr(x)}
where `p`, `q`, and `r` are any of the
letters `s`, `c`, `d`, `n`, with
the understanding that `ss=cc=dd=nn=1`.
Let
.. math::
u=\int_0^\phi \frac{d\theta} {\sqrt {1-m \sin^2 \theta}}
Then the *Jacobi elliptic function* `sn(u)` is given by
.. math::
{sn}\; u = \sin \phi
and `cn(u)` is given by
.. math::
{cn}\; u = \cos \phi
and
.. math::
{dn}\; u = \sqrt {1-m\sin^2 \phi}.
To emphasize the dependence on `m`, one can write
`sn(u,m)` for example (and similarly for `cn` and
`dn`). This is the notation used below.
For a given `k` with `0 < k < 1` they therefore are
solutions to the following nonlinear ordinary differential
equations:
- `\mathrm{sn}\,(x;k)` solves the differential equations
.. math::
\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1+k^2) y - 2 k^2 y^3 = 0,
and
.. math::
\left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-k^2 y^2).
- `\mathrm{cn}\,(x;k)` solves the differential equations
.. math::
\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1-2k^2) y + 2 k^2 y^3 = 0,
and `\left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-k^2 + k^2 y^2)`.
- `\mathrm{dn}\,(x;k)` solves the differential equations
.. math::
\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - (2 - k^2) y + 2 y^3 = 0,
and `\left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2= y^2 (1 - k^2 - y^2)`.
If `K(m)` denotes the complete elliptic integral of the
first kind (denoted ``elliptic_kc``), the elliptic functions
`sn (x,m)` and `cn (x,m)` have real periods
`4K(m)`, whereas `dn (x,m)` has a period
`2K(m)`. The limit `m\rightarrow 0` gives
`K(0) = \pi/2` and trigonometric functions:
`sn(x, 0) = \sin x`, `cn(x, 0) = \cos x`,
`dn(x, 0) = 1`. The limit `m \rightarrow 1` gives
`K(1) \rightarrow \infty` and hyperbolic functions:
`sn(x, 1) = \tanh x`,
`cn(x, 1) = \mbox{\rm sech} x`,
`dn(x, 1) = \mbox{\rm sech} x`.
- The incomplete elliptic integrals (of the first kind, etc.) are:
.. math::
\begin{array}{c} \displaystyle\int_0^\phi \frac{1}{\sqrt{1 - m\sin(x)^2}}\, dx,\\ \displaystyle\int_0^\phi \sqrt{1 - m\sin(x)^2}\, dx,\\ \displaystyle\int_0^\phi \frac{\sqrt{1-mt^2}}{\sqrt(1 - t^2)}\, dx,\\ \displaystyle\int_0^\phi \frac{1}{\sqrt{1 - m\sin(x)^2\sqrt{1 - n\sin(x)^2}}}\, dx, \end{array}
and the complete ones are obtained by taking `\phi =\pi/2`.
REFERENCES:
- Abramowitz and Stegun: Handbook of Mathematical Functions,
http://www.math.sfu.ca/~cbm/aands/
- http://en.wikipedia.org/wiki/Bessel_function
- http://en.wikipedia.org/wiki/Airy_function
- http://en.wikipedia.org/wiki/Spherical_harmonics
- http://en.wikipedia.org/wiki/Helmholtz_equation
- http://en.wikipedia.org/wiki/Jacobi's_elliptic_functions
- A. Khare, U. Sukhatme, 'Cyclic Identities Involving
Jacobi Elliptic Functions', Math ArXiv, math-ph/0201004
- Online Encyclopedia of Special Function
http://algo.inria.fr/esf/index.html
TODO: Resolve weird bug in commented out code in hypergeometric_U
below.
AUTHORS:
- David Joyner and William Stein
Added 16-02-2008 (wdj): optional calls to scipy and replace all
'#random' by '...' (both at the request of William Stein)
.. warning::
SciPy's versions are poorly documented and seem less
accurate than the Maxima and PARI versions.
"""
#*****************************************************************************
# Copyright (C) 2006 William Stein <wstein@gmail.com>
# 2006 David Joyner <wdj@usna.edu>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# This code is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# The full text of the GPL is available at:
#
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.plot.plot import plot
from sage.rings.real_mpfr import RealField
from sage.rings.complex_field import ComplexField
from sage.misc.sage_eval import sage_eval
from sage.rings.all import ZZ, RR, RDF
from sage.functions.other import real, imag, log_gamma
from sage.symbolic.function import BuiltinFunction
from sage.calculus.calculus import maxima
_done = False
def _init():
"""
Internal function which checks if Maxima has loaded the
"orthopoly" package. All functions using this in this
file should call this function first.
TEST:
The global starts ``False``::
sage: sage.functions.special._done
False
Then after using one of these functions, it changes::
sage: from sage.functions.special import airy_ai
sage: airy_ai(1.0)
0.135292416313
sage: sage.functions.special._done
True
"""
global _done
if _done:
return
maxima.eval('load("orthopoly");')
maxima.eval('orthopoly_returns_intervals:false;')
_done = True
def meval(x):
"""
Returns ``x`` evaluated in Maxima, then returned to Sage.
This is used to evaluate several of these special functions.
TEST::
sage: from sage.functions.special import airy_ai
sage: airy_bi(1.0)
1.20742359495
"""
return maxima(x).sage()
class MaximaFunction(BuiltinFunction):
"""
EXAMPLES::
sage: from sage.functions.special import MaximaFunction
sage: f = MaximaFunction("jacobi_sn")
sage: f(1,1)
tanh(1)
sage: f(1/2,1/2).n()
0.470750473655657
"""
def __init__(self, name, nargs=2, conversions={}):
"""
EXAMPLES::
sage: from sage.functions.special import MaximaFunction
sage: f = MaximaFunction("jacobi_sn")
sage: f(1,1)
tanh(1)
sage: f(1/2,1/2).n()
0.470750473655657
"""
c = dict(maxima=name)
c.update(conversions)
BuiltinFunction.__init__(self, name=name, nargs=nargs,
conversions=c)
def _maxima_init_evaled_(self, *args):
"""
Returns a string which represents this function evaluated at
*args* in Maxima.
EXAMPLES::
sage: from sage.functions.special import MaximaFunction
sage: f = MaximaFunction("jacobi_sn")
sage: f._maxima_init_evaled_(1/2, 1/2)
'jacobi_sn(1/2, 1/2)'
"""
args_maxima = []
for a in args:
if isinstance(a, str):
args_maxima.append(a)
elif hasattr(a, '_maxima_init_'):
args_maxima.append(a._maxima_init_())
else:
args_maxima.append(str(a))
return "%s(%s)"%(self.name(), ', '.join(args_maxima))
def _evalf_(self, *args, **kwds):
"""
Returns a numerical approximation of this function using
Maxima. Currently, this is limited to 53 bits of precision.
EXAMPLES::
sage: from sage.functions.special import MaximaFunction
sage: f = MaximaFunction("jacobi_sn")
sage: f(1/2,1/2)
jacobi_sn(1/2, 1/2)
sage: f(1/2,1/2).n()
0.470750473655657
TESTS::
sage: f(1/2,1/2).n(150)
Traceback (most recent call last):
...
NotImplementedError: jacobi_sn not implemented for precision > 53
"""
parent = kwds['parent']
if hasattr(parent, 'prec') and parent.prec() > 53:
raise NotImplementedError, "%s not implemented for precision > 53"%self.name()
_init()
return parent(maxima("%s, numer"%self._maxima_init_evaled_(*args)))
def _eval_(self, *args):
"""
Returns a string which represents this function evaluated at
*args* in Maxima.
EXAMPLES::
sage: from sage.functions.special import MaximaFunction
sage: f = MaximaFunction("jacobi_sn")
sage: f(1,1)
tanh(1)
sage: f._eval_(1,1)
tanh(1)
Here arccoth doesn't have 1 in its domain, so we just hold the expression:
sage: elliptic_e(arccoth(1), x^2*e)
elliptic_e(arccoth(1), x^2*e)
"""
_init()
try:
s = maxima(self._maxima_init_evaled_(*args))
except TypeError:
return None
if self.name() in repr(s):
return None
else:
return s.sage()
from sage.misc.cachefunc import cached_function
@cached_function
def maxima_function(name):
"""
Returns a function which is evaluated both symbolically and
numerically via Maxima. In particular, it returns an instance
of :class:`MaximaFunction`.
.. note::
This function is cached so that duplicate copies of the same
function are not created.
EXAMPLES::
sage: n(bessel_J(3,10,"maxima"))
0.0583793793051...
sage: spherical_hankel2(2,i)
-e
"""
# The superclass of MaximaFunction, BuiltinFunction, assumes that there
# will be only one symbolic function with the same name and class.
# We create a new class for each Maxima function wrapped.
class NewMaximaFunction(MaximaFunction):
def __init__(self):
"""
Constructs an object that wraps a Maxima function.
TESTS::
sage: n(bessel_J(3,10,"maxima"))
0.0583793793051...
sage: spherical_hankel2(2,x)
(-I*x^2 - 3*x + 3*I)*e^(-I*x)/x^3
"""
MaximaFunction.__init__(self, name)
return NewMaximaFunction()
def airy_ai(x):
r"""
The function `Ai(x)` and the related function `Bi(x)`,
which is also called an *Airy function*, are
solutions to the differential equation
.. math::
y'' - xy = 0,
known as the *Airy equation*. The initial conditions
`Ai(0) = (\Gamma(2/3)3^{2/3})^{-1}`,
`Ai'(0) = -(\Gamma(1/3)3^{1/3})^{-1}` define `Ai(x)`.
The initial conditions `Bi(0) = 3^{1/2}Ai(0)`,
`Bi'(0) = -3^{1/2}Ai'(0)` define `Bi(x)`.
They are named after the British astronomer George Biddell Airy.
They belong to the class of "Bessel functions of fractional order".
EXAMPLES::
sage: airy_ai(1.0) # last few digits are random
0.135292416312881400
sage: airy_bi(1.0) # last few digits are random
1.20742359495287099
REFERENCE:
- Abramowitz and Stegun: Handbook of Mathematical Functions,
http://www.math.sfu.ca/~cbm/aands/
- http://en.wikipedia.org/wiki/Airy_function
"""
_init()
return RDF(meval("airy_ai(%s)"%RDF(x)))
def airy_bi(x):
r"""
The function `Ai(x)` and the related function `Bi(x)`,
which is also called an *Airy function*, are
solutions to the differential equation
.. math::
y'' - xy = 0,
known as the *Airy equation*. The initial conditions
`Ai(0) = (\Gamma(2/3)3^{2/3})^{-1}`,
`Ai'(0) = -(\Gamma(1/3)3^{1/3})^{-1}` define `Ai(x)`.
The initial conditions `Bi(0) = 3^{1/2}Ai(0)`,
`Bi'(0) = -3^{1/2}Ai'(0)` define `Bi(x)`.
They are named after the British astronomer George Biddell Airy.
They belong to the class of "Bessel functions of fractional order".
EXAMPLES::
sage: airy_ai(1) # last few digits are random
0.135292416312881400
sage: airy_bi(1) # last few digits are random
1.20742359495287099
REFERENCE:
- Abramowitz and Stegun: Handbook of Mathematical Functions,
http://www.math.sfu.ca/~cbm/aands/
- http://en.wikipedia.org/wiki/Airy_function
"""
_init()
return RDF(meval("airy_bi(%s)"%RDF(x)))
def bessel_I(nu,z,algorithm = "pari",prec=53):
r"""
Implements the "I-Bessel function", or "modified Bessel function,
1st kind", with index (or "order") nu and argument z.
INPUT:
- ``nu`` - a real (or complex, for pari) number
- ``z`` - a real (positive) algorithm - "pari" or
"maxima" or "scipy" prec - real precision (for PARI only)
DEFINITION::
Maxima:
inf
==== - nu - 2 k nu + 2 k
\ 2 z
> -------------------
/ k! Gamma(nu + k + 1)
====
k = 0
PARI:
inf
==== - 2 k 2 k
\ 2 z Gamma(nu + 1)
> -----------------------
/ k! Gamma(nu + k + 1)
====
k = 0
Sometimes ``bessel_I(nu,z)`` is denoted
``I_nu(z)`` in the literature.
.. warning::
In Maxima (the manual says) i0 is deprecated but
``bessel_i(0,*)`` is broken. (Was fixed in recent CVS patch
though.)
EXAMPLES::
sage: bessel_I(1,1,"pari",500)
0.565159103992485027207696027609863307328899621621092009480294489479255640964371134092664997766814410064677886055526302676857637684917179812041131208121
sage: bessel_I(1,1)
0.565159103992485
sage: bessel_I(2,1.1,"maxima")
0.16708949925104...
sage: bessel_I(0,1.1,"maxima")
1.32616018371265...
sage: bessel_I(0,1,"maxima")
1.2660658777520...
sage: bessel_I(1,1,"scipy")
0.565159103992...
Check whether the return value is real whenever the argument is real (#10251)::
sage: bessel_I(5, 1.5, algorithm='scipy') in RR
True
"""
if algorithm=="pari":
from sage.libs.pari.all import pari
try:
R = RealField(prec)
nu = R(nu)
z = R(z)
except TypeError:
C = ComplexField(prec)
nu = C(nu)
z = C(z)
K = C
K = z.parent()
return K(pari(nu).besseli(z, precision=prec))
elif algorithm=="scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.iv(float(nu),complex(real(z),imag(z))))
ans = ans.replace("(","")
ans = ans.replace(")","")
ans = ans.replace("j","*I")
ans = sage_eval(ans)
return real(ans) if z in RR else ans # Return real value when arg is real
elif algorithm == "maxima":
if prec != 53:
raise ValueError, "for the maxima algorithm the precision must be 53"
return sage_eval(maxima.eval("bessel_i(%s,%s)"%(float(nu),float(z))))
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def bessel_J(nu,z,algorithm="pari",prec=53):
r"""
Return value of the "J-Bessel function", or "Bessel function, 1st
kind", with index (or "order") nu and argument z.
::
Defn:
Maxima:
inf
==== - nu - 2 k nu + 2 k
\ (-1)^k 2 z
> -------------------------
/ k! Gamma(nu + k + 1)
====
k = 0
PARI:
inf
==== - 2k 2k
\ (-1)^k 2 z Gamma(nu + 1)
> ----------------------------
/ k! Gamma(nu + k + 1)
====
k = 0
Sometimes bessel_J(nu,z) is denoted J_nu(z) in the literature.
.. warning::
Inaccurate for small values of z.
EXAMPLES::
sage: bessel_J(2,1.1)
0.136564153956658
sage: bessel_J(0,1.1)
0.719622018527511
sage: bessel_J(0,1)
0.765197686557967
sage: bessel_J(0,0)
1.00000000000000
sage: bessel_J(0.1,0.1)
0.777264368097005
We check consistency of PARI and Maxima::
sage: n(bessel_J(3,10,"maxima"))
0.0583793793051...
sage: n(bessel_J(3,10,"pari"))
0.0583793793051868
sage: bessel_J(3,10,"scipy")
0.0583793793052...
Check whether the return value is real whenever the argument is real (#10251)::
sage: bessel_J(5, 1.5, algorithm='scipy') in RR
True
"""
if algorithm=="pari":
from sage.libs.pari.all import pari
try:
R = RealField(prec)
nu = R(nu)
z = R(z)
except TypeError:
C = ComplexField(prec)
nu = C(nu)
z = C(z)
K = C
if nu == 0:
nu = ZZ(0)
K = z.parent()
return K(pari(nu).besselj(z, precision=prec))
elif algorithm=="scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.jv(float(nu),complex(real(z),imag(z))))
ans = ans.replace("(","")
ans = ans.replace(")","")
ans = ans.replace("j","*I")
ans = sage_eval(ans)
return real(ans) if z in RR else ans
elif algorithm == "maxima":
if prec != 53:
raise ValueError, "for the maxima algorithm the precision must be 53"
return maxima_function("bessel_j")(nu, z)
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def bessel_K(nu,z,algorithm="pari",prec=53):
r"""
Implements the "K-Bessel function", or "modified Bessel function,
2nd kind", with index (or "order") nu and argument z. Defn::
pi*(bessel_I(-nu, z) - bessel_I(nu, z))
----------------------------------------
2*sin(pi*nu)
if nu is not an integer and by taking a limit otherwise.
Sometimes bessel_K(nu,z) is denoted K_nu(z) in the literature. In
PARI, nu can be complex and z must be real and positive.
EXAMPLES::
sage: bessel_K(3,2,"scipy")
0.64738539094...
sage: bessel_K(3,2)
0.64738539094...
sage: bessel_K(1,1)
0.60190723019...
sage: bessel_K(1,1,"pari",10)
0.60
sage: bessel_K(1,1,"pari",100)
0.60190723019723457473754000154
TESTS::
sage: bessel_K(2,1.1, algorithm="maxima")
Traceback (most recent call last):
...
NotImplementedError: The K-Bessel function is only implemented for the pari and scipy algorithms
Check whether the return value is real whenever the argument is real (#10251)::
sage: bessel_K(5, 1.5, algorithm='scipy') in RR
True
"""
if algorithm=="scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.kv(float(nu),float(z)))
ans = ans.replace("(","")
ans = ans.replace(")","")
ans = ans.replace("j","*I")
ans = sage_eval(ans)
return real(ans) if z in RR else ans
elif algorithm == 'pari':
from sage.libs.pari.all import pari
try:
R = RealField(prec)
nu = R(nu)
z = R(z)
except TypeError:
C = ComplexField(prec)
nu = C(nu)
z = C(z)
K = C
K = z.parent()
return K(pari(nu).besselk(z, precision=prec))
elif algorithm == 'maxima':
raise NotImplementedError, "The K-Bessel function is only implemented for the pari and scipy algorithms"
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def bessel_Y(nu,z,algorithm="maxima", prec=53):
r"""
Implements the "Y-Bessel function", or "Bessel function of the 2nd
kind", with index (or "order") nu and argument z.
.. note::
Currently only prec=53 is supported.
Defn::
cos(pi n)*bessel_J(nu, z) - bessel_J(-nu, z)
-------------------------------------------------
sin(nu*pi)
if nu is not an integer and by taking a limit otherwise.
Sometimes bessel_Y(n,z) is denoted Y_n(z) in the literature.
This is computed using Maxima by default.
EXAMPLES::
sage: bessel_Y(2,1.1,"scipy")
-1.4314714939...
sage: bessel_Y(2,1.1)
-1.4314714939590...
sage: bessel_Y(3.001,2.1)
-1.0299574976424...
TESTS::
sage: bessel_Y(2,1.1, algorithm="pari")
Traceback (most recent call last):
...
NotImplementedError: The Y-Bessel function is only implemented for the maxima and scipy algorithms
"""
if algorithm=="scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.yv(float(nu),complex(real(z),imag(z))))
ans = ans.replace("(","")
ans = ans.replace(")","")
ans = ans.replace("j","*I")
ans = sage_eval(ans)
return real(ans) if z in RR else ans
elif algorithm == "maxima":
if prec != 53:
raise ValueError, "for the maxima algorithm the precision must be 53"
return RR(maxima.eval("bessel_y(%s,%s)"%(float(nu),float(z))))
elif algorithm == "pari":
raise NotImplementedError, "The Y-Bessel function is only implemented for the maxima and scipy algorithms"
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
class Bessel():
"""
A class implementing the I, J, K, and Y Bessel functions.
EXAMPLES::
sage: g = Bessel(2); g
J_{2}
sage: print g
J-Bessel function of order 2
sage: g.plot(0,10)
::
sage: Bessel(2, typ='I')(pi)
2.61849485263445
sage: Bessel(2, typ='J')(pi)
0.485433932631509
sage: Bessel(2, typ='K')(pi)
0.0510986902537926
sage: Bessel(2, typ='Y')(pi)
-0.0999007139289404
"""
def __init__(self, nu, typ = "J", algorithm = None, prec = 53):
"""
Initializes new instance of the Bessel class.
INPUT:
- ``typ`` -- (default: J) the type of Bessel function: 'I', 'J', 'K'
or 'Y'.
- ``algorithm`` -- (default: maxima for type Y, pari for other types)
algorithm to use to compute the Bessel function: 'pari', 'maxima' or
'scipy'. Note that type K is not implemented in Maxima and type Y
is not implemented in PARI.
- ``prec`` -- (default: 53) precision in bits of the Bessel function.
Only supported for the PARI algorithm.
EXAMPLES::
sage: g = Bessel(2); g
J_{2}
sage: Bessel(1,'I')
I_{1}
sage: Bessel(6, prec=120)(pi)
0.014545966982505560573660369604001804
sage: Bessel(6, algorithm="pari")(pi)
0.0145459669825056
For the Bessel J-function, Maxima returns a symbolic result. For
types I and Y, we always get a numeric result::
sage: b = Bessel(6, algorithm="maxima")(pi); b
bessel_j(6, pi)
sage: b.n(53)
0.0145459669825056
sage: Bessel(6, typ='I', algorithm="maxima")(pi)
0.0294619840059568
sage: Bessel(6, typ='Y', algorithm="maxima")(pi)
-4.33932818939038
SciPy usually gives less precise results::
sage: Bessel(6, algorithm="scipy")(pi)
0.0145459669825000...
TESTS::
sage: Bessel(1,'Z')
Traceback (most recent call last):
...
ValueError: typ must be one of I, J, K, Y
"""
if not (typ in ['I', 'J', 'K', 'Y']):
raise ValueError, "typ must be one of I, J, K, Y"
# Did the user ask for the default algorithm?
if algorithm is None:
if typ == 'Y':
algorithm = 'maxima'
else:
algorithm = 'pari'
self._system = algorithm
self._order = nu
self._type = typ
prec = int(prec)
if prec < 0:
raise ValueError, "prec must be a positive integer"
self._prec = int(prec)
def __str__(self):
"""
Returns a string representation of this Bessel object.
TEST::
sage: a = Bessel(1,'I')
sage: str(a)
'I-Bessel function of order 1'
"""
return self.type()+"-Bessel function of order "+str(self.order())
def __repr__(self):
"""
Returns a string representation of this Bessel object.
TESTS::
sage: Bessel(1,'I')
I_{1}
"""
return self.type()+"_{"+str(self.order())+"}"
def type(self):
"""
Returns the type of this Bessel object.
TEST::
sage: a = Bessel(3,'K')
sage: a.type()
'K'
"""
return self._type
def prec(self):
"""
Returns the precision (in number of bits) used to represent this
Bessel function.
TESTS::
sage: a = Bessel(3,'K')
sage: a.prec()
53
sage: B = Bessel(20,prec=100); B
J_{20}
sage: B.prec()
100
"""
return self._prec
def order(self):
"""
Returns the order of this Bessel function.
TEST::
sage: a = Bessel(3,'K')
sage: a.order()
3
"""
return self._order
def system(self):
"""
Returns the package used, e.g. Maxima, PARI, or SciPy, to compute with
this Bessel function.
TESTS::
sage: Bessel(20,algorithm='maxima').system()
'maxima'
sage: Bessel(20,prec=100).system()
'pari'
"""
return self._system
def __call__(self,z):
"""
Implements evaluation of all the Bessel functions directly
from the Bessel class. This essentially allows a Bessel object to
behave like a function that can be invoked.
TESTS::
sage: Bessel(3,'K')(5.0)
0.00829176841523093
sage: Bessel(20,algorithm='maxima')(5.0)
2.77033005213e-11
sage: Bessel(20,prec=100)(5.0101010101010101)
2.8809188227195382093062257967e-11
sage: B = Bessel(2,'Y',algorithm='scipy',prec=50)
sage: B(2.0)
Traceback (most recent call last):
...
ValueError: for the scipy algorithm the precision must be 53
"""
nu = self.order()
t = self.type()
s = self.system()
p = self.prec()
if t == "I":
return bessel_I(nu,z,algorithm=s,prec=p)
if t == "J":
return bessel_J(nu,z,algorithm=s,prec=p)
if t == "K":
return bessel_K(nu,z,algorithm=s,prec=p)
if t == "Y":
return bessel_Y(nu,z,algorithm=s,prec=p)
def plot(self,a,b):
"""
Enables easy plotting of all the Bessel functions directly
from the Bessel class.
TESTS::
sage: plot(Bessel(2),3,4)
sage: Bessel(2).plot(3,4)
sage: P = Bessel(2,'I').plot(1,5)
sage: P += Bessel(2,'J').plot(1,5)
sage: P += Bessel(2,'K').plot(1,5)
sage: P += Bessel(2,'Y').plot(1,5)
sage: show(P)
"""
nu = self.order()
s = self.system()
t = self.type()
if t == "I":
f = lambda z: bessel_I(nu,z,s)
P = plot(f,a,b)
if t == "J":
f = lambda z: bessel_J(nu,z,s)
P = plot(f,a,b)
if t == "K":
f = lambda z: bessel_K(nu,z,s)
P = plot(f,a,b)
if t == "Y":
f = lambda z: bessel_Y(nu,z,s)
P = plot(f,a,b)
return P
def hypergeometric_U(alpha,beta,x,algorithm="pari",prec=53):
r"""
Default is a wrap of PARI's hyperu(alpha,beta,x) function.
Optionally, algorithm = "scipy" can be used.
The confluent hypergeometric function `y = U(a,b,x)` is
defined to be the solution to Kummer's differential equation
.. math::
xy'' + (b-x)y' - ay = 0.
This satisfies `U(a,b,x) \sim x^{-a}`, as
`x\rightarrow \infty`, and is sometimes denoted
``x^{-a}2_F_0(a,1+a-b,-1/x)``. This is not the same as Kummer's
`M`-hypergeometric function, denoted sometimes as
``_1F_1(alpha,beta,x)``, though it satisfies the same DE that
`U` does.
.. warning::
In the literature, both are called "Kummer confluent
hypergeometric" functions.
EXAMPLES::
sage: hypergeometric_U(1,1,1,"scipy")
0.596347362323...
sage: hypergeometric_U(1,1,1)
0.59634736232319...
sage: hypergeometric_U(1,1,1,"pari",70)
0.59634736232319407434...
"""
if algorithm=="scipy":
if prec != 53:
raise ValueError, "for the scipy algorithm the precision must be 53"
import scipy.special
ans = str(scipy.special.hyperu(float(alpha),float(beta),float(x)))
ans = ans.replace("(","")
ans = ans.replace(")","")
ans = ans.replace("j","*I")
return sage_eval(ans)
elif algorithm=='pari':
from sage.libs.pari.all import pari
R = RealField(prec)
return R(pari(R(alpha)).hyperu(R(beta), R(x), precision=prec))
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def spherical_bessel_J(n, var, algorithm="maxima"):
r"""
Returns the spherical Bessel function of the first kind for
integers n >= 1.
Reference: AS 10.1.8 page 437 and AS 10.1.15 page 439.
EXAMPLES::
sage: spherical_bessel_J(2,x)
((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x
sage: spherical_bessel_J(1, 5.2, algorithm='scipy')
-0.12277149950007...
sage: spherical_bessel_J(1, 3, algorithm='scipy')
0.345677499762355...
"""
if algorithm=="scipy":
from scipy.special.specfun import sphj
return sphj(int(n), float(var))[1][-1]
elif algorithm == 'maxima':
_init()
return meval("spherical_bessel_j(%s,%s)"%(ZZ(n),var))
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def spherical_bessel_Y(n,var, algorithm="maxima"):
r"""
Returns the spherical Bessel function of the second kind for
integers n -1.
Reference: AS 10.1.9 page 437 and AS 10.1.15 page 439.
EXAMPLES::
sage: x = PolynomialRing(QQ, 'x').gen()
sage: spherical_bessel_Y(2,x)
-((3/x^2 - 1)*cos(x) + 3*sin(x)/x)/x
"""
if algorithm=="scipy":
import scipy.special
ans = str(scipy.special.sph_yn(int(n),float(var)))
ans = ans.replace("(","")
ans = ans.replace(")","")
ans = ans.replace("j","*I")
return sage_eval(ans)
elif algorithm == 'maxima':
_init()
return meval("spherical_bessel_y(%s,%s)"%(ZZ(n),var))
else:
raise ValueError, "unknown algorithm '%s'"%algorithm
def spherical_hankel1(n, var):
r"""
Returns the spherical Hankel function of the first kind for
integers `n > -1`, written as a string. Reference: AS
10.1.36 page 439.
EXAMPLES::
sage: spherical_hankel1(2, x)
(I*x^2 - 3*x - 3*I)*e^(I*x)/x^3
"""
return maxima_function("spherical_hankel1")(ZZ(n), var)
def spherical_hankel2(n,x):
r"""
Returns the spherical Hankel function of the second kind for
integers `n > -1`, written as a string. Reference: AS 10.1.17 page
439.
EXAMPLES::
sage: spherical_hankel2(2, x)
(-I*x^2 - 3*x + 3*I)*e^(-I*x)/x^3
Here I = sqrt(-1).
"""
return maxima_function("spherical_hankel2")(ZZ(n), x)
def spherical_harmonic(m,n,x,y):
r"""
Returns the spherical Harmonic function of the second kind for
integers `n > -1`, `|m|\leq n`. Reference:
Merzbacher 9.64.
EXAMPLES::
sage: x,y = var('x,y')
sage: spherical_harmonic(3,2,x,y)
15/4*sqrt(7/30)*e^(2*I*y)*sin(x)^2*cos(x)/sqrt(pi)
sage: spherical_harmonic(3,2,1,2)
15/4*sqrt(7/30)*e^(4*I)*sin(1)^2*cos(1)/sqrt(pi)
"""
_init()
return meval("spherical_harmonic(%s,%s,%s,%s)"%(ZZ(m),ZZ(n),x,y))
####### elliptic functions and integrals
def elliptic_j(z):
r"""
Returns the elliptic modular `j`-function evaluated at `z`.
INPUT:
- ``z`` (complex) -- a complex number with positive imaginary part.
OUTPUT:
(complex) The value of `j(z)`.
ALGORITHM:
Calls the ``pari`` function ``ellj()``.
AUTHOR:
John Cremona
EXAMPLES::
sage: elliptic_j(CC(i))
1728.00000000000
sage: elliptic_j(sqrt(-2.0))
8000.00000000000
sage: z = ComplexField(100)(1,sqrt(11))/2
sage: elliptic_j(z)
-32768.000...
sage: elliptic_j(z).real().round()
-32768
"""
CC = z.parent()
from sage.rings.complex_field import is_ComplexField
if not is_ComplexField(CC):
CC = ComplexField()
try:
z = CC(z)
except ValueError:
raise ValueError, "elliptic_j only defined for complex arguments."
from sage.libs.all import pari
return CC(pari(z).ellj())
def jacobi(sym,x,m):
r"""
Here sym = "pq", where p,q in c,d,n,s. This returns the value of
the Jacobi function pq(x,m), as described in the documentation for
Sage's "special" module. There are a total of 12 functions
described by this.
EXAMPLES::
sage: jacobi("sn",1,1)
tanh(1)
sage: jacobi("cd",1,1/2)
jacobi_cd(1, 1/2)
sage: RDF(jacobi("cd",1,1/2))
0.724009721659
sage: RDF(jacobi("cn",1,1/2)); RDF(jacobi("dn",1,1/2)); RDF(jacobi("cn",1,1/2)/jacobi("dn",1,1/2))
0.595976567672
0.823161001632
0.724009721659
sage: jsn = jacobi("sn",x,1)
sage: P = plot(jsn,0,1)
To view this, type P.show().
"""
return maxima_function("jacobi_%s"%sym)(x,m)
def inverse_jacobi(sym,x,m):
"""
Here sym = "pq", where p,q in c,d,n,s. This returns the value of
the inverse Jacobi function `pq^{-1}(x,m)`. There are a
total of 12 functions described by this.
EXAMPLES::
sage: jacobi("sn",1/2,1/2)
jacobi_sn(1/2, 1/2)
sage: float(jacobi("sn",1/2,1/2))
0.4707504736556572
sage: float(inverse_jacobi("sn",0.47,1/2))
0.4990982313222196
sage: float(inverse_jacobi("sn",0.4707504,0.5))
0.4999999114665548
sage: P = plot(inverse_jacobi('sn', x, 0.5), 0, 1, plot_points=20)
Now to view this, just type show(P).
"""
return maxima_function("inverse_jacobi_%s"%sym)(x,m)
#### elliptic integrals
class EllipticE(MaximaFunction):
r"""
This returns the value of the "incomplete elliptic integral of the
second kind,"
.. math::
\int_0^\phi \sqrt{1 - m\sin(x)^2}\, dx,
i.e., ``integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)``. Taking `\phi
= \pi/2` gives ``elliptic_ec``.
EXAMPLES::
sage: z = var("z")
sage: # this is still wrong: must be abs(sin(z)) + 2*round(z/pi)
sage: elliptic_e(z, 1)
sin(z) + 2*round(z/pi)
sage: elliptic_e(z, 0)
z
sage: elliptic_e(0.5, 0.1)
0.498011394499
sage: loads(dumps(elliptic_e))
elliptic_e
"""
def __init__(self):
"""
EXAMPLES::
sage: elliptic_e(0.5, 0.1)
0.498011394499
"""
MaximaFunction.__init__(self, "elliptic_e")
elliptic_e = EllipticE()
class EllipticEC(MaximaFunction):
"""
This returns the value of the "complete elliptic integral of the
second kind,"
.. math::
\int_0^{\pi/2} \sqrt{1 - m\sin(x)^2}\, dx.
EXAMPLES::
sage: elliptic_ec(0.1)
1.5307576369
sage: elliptic_ec(x).diff()
1/2*(elliptic_ec(x) - elliptic_kc(x))/x
sage: loads(dumps(elliptic_ec))
elliptic_ec
"""
def __init__(self):
"""
EXAMPLES::
sage: elliptic_ec(0.1)
1.5307576369
"""
MaximaFunction.__init__(self, "elliptic_ec", nargs=1)
def _derivative_(self, *args, **kwds):
"""
EXAMPLES::
sage: elliptic_ec(x).diff()
1/2*(elliptic_ec(x) - elliptic_kc(x))/x
"""
diff_param = kwds['diff_param']
assert diff_param == 0
x = args[diff_param]
return (elliptic_ec(x) - elliptic_kc(x))/(2*x)
elliptic_ec = EllipticEC()
class EllipticEU(MaximaFunction):
r"""
This returns the value of the "incomplete elliptic integral of the
second kind,"
.. math::
\int_0^u \mathrm{dn}(x,m)^2\, dx = \int_0^\tau
{\sqrt{1-m x^2}\over\sqrt{1-x^2}}\, dx.
where `\tau = \mathrm{sn}(u, m)`.
EXAMPLES::
sage: elliptic_eu (0.5, 0.1)
0.496054551287
"""
def __init__(self):
r"""
EXAMPLES::
sage: elliptic_eu (0.5, 0.1)
0.496054551287
"""
MaximaFunction.__init__(self, "elliptic_eu")
elliptic_eu = EllipticEU()
class EllipticF(MaximaFunction):
r"""
This returns the value of the "incomplete elliptic integral of the
first kind,"
.. math::
\int_0^\phi \frac{dx}{\sqrt{1 - m\sin(x)^2}},
i.e., ``integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)``. Taking
`\phi = \pi/2` gives ``elliptic_kc``.
EXAMPLES::
sage: z = var("z")
sage: elliptic_f (z, 0)
z
sage: elliptic_f (z, 1)
log(tan(1/4*pi + 1/2*z))
sage: elliptic_f (0.2, 0.1)
0.200132506748
"""
def __init__(self):
r"""
EXAMPLES::
sage: elliptic_f (0.2, 0.1)
0.200132506748
"""
MaximaFunction.__init__(self, "elliptic_f")
elliptic_f = EllipticF()
class EllipticKC(MaximaFunction):
r"""
This returns the value of the "complete elliptic integral of the
first kind,"
.. math::
\int_0^{\pi/2} \frac{dx}{\sqrt{1 - m\sin(x)^2}}.
EXAMPLES::
sage: elliptic_kc(0.5)
1.8540746773
sage: elliptic_f(RR(pi/2), 0.5)
1.8540746773
"""
def __init__(self):
r"""
EXAMPLES::
sage: elliptic_kc(0.5)
1.8540746773
sage: elliptic_f(RR(pi/2), 0.5)
1.8540746773
"""
MaximaFunction.__init__(self, "elliptic_kc", nargs=1)
elliptic_kc = EllipticKC()
class EllipticPi(MaximaFunction):
r"""
This returns the value of the "incomplete elliptic integral of the
third kind,"
.. math::
\text{elliptic\_pi}(n, t, m) = \int_0^t \frac{dx}{(1 - n \sin(x)^2)
\sqrt{1 - m \sin(x)^2}}.
INPUT:
- ``n`` -- a real number, called the "characteristic"
- ``t`` -- a real number, called the "amplitude"
- ``m`` -- a real number, called the "parameter"
EXAMPLES::
sage: N(elliptic_pi(1, pi/4, 1))
1.14779357469632
Compare the value computed by Maxima to the definition as a definite integral
(using GSL)::
sage: elliptic_pi(0.1, 0.2, 0.3)
0.200665068221
sage: numerical_integral(1/(1-0.1*sin(x)^2)/sqrt(1-0.3*sin(x)^2), 0.0, 0.2)
(0.2006650682209791, 2.227829789769088e-15)
ALGORITHM:
Numerical evaluation and symbolic manipulation are provided by `Maxima`_.
REFERENCES:
- Abramowitz and Stegun: Handbook of Mathematical Functions, section 17.7
http://www.math.sfu.ca/~cbm/aands/
- Elliptic Functions in `Maxima`_
.. _`Maxima`: http://maxima.sourceforge.net/docs/manual/en/maxima_16.html#SEC91
"""
def __init__(self):
r"""
EXAMPLES::
sage: elliptic_pi(0.1, 0.2, 0.3)
0.200665068221
"""
MaximaFunction.__init__(self, "elliptic_pi", nargs=3)
elliptic_pi = EllipticPi()
def lngamma(t):
r"""
This method is deprecated, please use
:meth:`~sage.functions.other.log_gamma` instead.
See the :meth:`~sage.functions.other.log_gamma` function for '
documentation and examples.
EXAMPLES::
sage: lngamma(RR(6))
doctest:...: DeprecationWarning: The method lngamma() is deprecated. Use log_gamma() instead.
4.78749174278205
"""
from sage.misc.misc import deprecation
deprecation("The method lngamma() is deprecated. Use log_gamma() instead.")
return log_gamma(t)
def exp_int(t):
r"""
The exponential integral `\int_t^\infty e^{-x}/x dx` (t
belongs to RR). This function is deprecated - please use
``Ei`` or ``exponential_integral_1`` as needed instead.
EXAMPLES::
sage: exp_int(6)
doctest:...: DeprecationWarning: The method exp_int() is deprecated. Use -Ei(-x) or exponential_integral_1(x) as needed instead.
0.000360082452162659
"""
from sage.misc.misc import deprecation
deprecation("The method exp_int() is deprecated. Use -Ei(-x) or exponential_integral_1(x) as needed instead.")
try:
return t.eint1()
except AttributeError:
from sage.libs.pari.all import pari
try:
return pari(t).eint1()
except:
raise NotImplementedError
def error_fcn(t):
r"""
The complementary error function
`\frac{2}{\sqrt{\pi}}\int_t^\infty e^{-x^2} dx` (t belongs
to RR). This function is currently always
evaluated immediately.
EXAMPLES::
sage: error_fcn(6)
2.15197367124989e-17
sage: error_fcn(RealField(100)(1/2))
0.47950012218695346231725334611
Note this is literally equal to `1 - erf(t)`::
sage: 1 - error_fcn(0.5)
0.520499877813047
sage: erf(0.5)
0.520499877813047
"""
try:
return t.erfc()
except AttributeError:
from sage.rings.real_mpfr import RR
try:
return RR(t).erfc()
except:
raise NotImplementedError