Coverage for local/lib/python2.7/site-packages/sage/functions/special.py : 59%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
r""" Miscellaneous 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
- Eviatar Bach (2013): making elliptic integrals symbolic
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.
Next, we summarize some of the properties of the functions implemented here.
- 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 ) = (-1)^m \sqrt{ \frac{(2\ell+1)}{4\pi} \frac{(\ell-m)!}{(\ell+m)!} } \, e^{i m \varphi } \, 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 .
- 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/
- :wikipedia:`Spherical_harmonics`
- :wikipedia:`Helmholtz_equation`
- Online Encyclopedia of Special Function http://algo.inria.fr/esf/index.html
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; typically they are limited by hardware floats precision. """
#***************************************************************************** # 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.rings.integer import Integer from sage.rings.real_mpfr import RealField from sage.rings.complex_field import ComplexField from sage.misc.latex import latex from sage.rings.all import ZZ, RR, RDF, CDF from .gamma import log_gamma from .other import real, imag from sage.symbolic.constants import pi from sage.symbolic.function import BuiltinFunction from sage.symbolic.expression import Expression from sage.calculus.calculus import maxima from sage.structure.element import parent from sage.libs.mpmath import utils as mpmath_utils from sage.functions.all import sqrt, sin, cot, exp from sage.symbolic.all import I
class SphericalHarmonic(BuiltinFunction): r""" Returns the spherical harmonic function `Y_n^m(\theta, \varphi)`.
For integers `n > -1`, `|m| \leq n`, simplification is done automatically. Numeric evaluation is supported for complex `n` and `m`.
EXAMPLES::
sage: x, y = var('x, y') sage: spherical_harmonic(3, 2, x, y) 1/8*sqrt(30)*sqrt(7)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi) sage: spherical_harmonic(3, 2, 1, 2) 1/8*sqrt(30)*sqrt(7)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi) sage: spherical_harmonic(3 + I, 2., 1, 2) -0.351154337307488 - 0.415562233975369*I sage: latex(spherical_harmonic(3, 2, x, y, hold=True)) Y_{3}^{2}\left(x, y\right) sage: spherical_harmonic(1, 2, x, y) 0 """ def __init__(self): r""" TESTS::
sage: n, m, theta, phi = var('n m theta phi') sage: spherical_harmonic(n, m, theta, phi)._sympy_() Ynm(n, m, theta, phi) """ BuiltinFunction.__init__(self, 'spherical_harmonic', nargs=4, conversions=dict( maple='SphericalY', mathematica= 'SphericalHarmonicY', maxima='spherical_harmonic', sympy='Ynm'))
def _eval_(self, n, m, theta, phi, **kwargs): r""" TESTS::
sage: x, y = var('x y') sage: spherical_harmonic(1, 2, x, y) 0 sage: spherical_harmonic(1, -2, x, y) 0 sage: spherical_harmonic(1/2, 2, x, y) spherical_harmonic(1/2, 2, x, y) sage: spherical_harmonic(3, 2, x, y) 1/8*sqrt(30)*sqrt(7)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi) sage: spherical_harmonic(3, 2, 1, 2) 1/8*sqrt(30)*sqrt(7)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi) sage: spherical_harmonic(3 + I, 2., 1, 2) -0.351154337307488 - 0.415562233975369*I
Check that :trac:`20939` is fixed::
sage: ex = spherical_harmonic(3,2,1,2*pi/3) sage: QQbar(ex * sqrt(pi)/cos(1)/sin(1)^2).minpoly() x^4 + 105/32*x^2 + 11025/1024 """ return sqrt((2*n+1)/4/pi) exp(I*m*phi) * gen_legendre_P(n, m, cos(theta)) * (-1)**m).simplify_trig()
def _evalf_(self, n, m, theta, phi, parent, **kwds): r""" TESTS::
sage: spherical_harmonic(3 + I, 2, 1, 2).n(100) -0.35115433730748836508201061672 - 0.41556223397536866209990358597*I sage: spherical_harmonic(I, I, I, I).n() 7.66678546069894 - 0.265754432549751*I """
def _derivative_(self, n, m, theta, phi, diff_param): r""" TESTS::
sage: n, m, theta, phi = var('n m theta phi') sage: spherical_harmonic(n, m, theta, phi).diff(theta) m*cot(theta)*spherical_harmonic(n, m, theta, phi) + sqrt(-(m + n + 1)*(m - n))*e^(-I*phi)*spherical_harmonic(n, m + 1, theta, phi) sage: spherical_harmonic(n, m, theta, phi).diff(phi) I*m*spherical_harmonic(n, m, theta, phi) """ sqrt((n - m) * (n + m + 1)) * exp(-I * phi) * spherical_harmonic(n, m + 1, theta, phi))
raise ValueError('only derivative with respect to theta or phi' ' supported')
def _latex_(self): r""" TESTS::
sage: latex(spherical_harmonic) Y_n^m """
def _print_latex_(self, n, m, theta, phi): r""" TESTS::
sage: y = var('y') sage: latex(spherical_harmonic(3, 2, x, y, hold=True)) Y_{3}^{2}\left(x, y\right) """ latex(n), latex(m), latex(theta), latex(phi))
spherical_harmonic = SphericalHarmonic()
####### 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
::
sage: tau = (1 + sqrt(-163))/2 sage: (-elliptic_j(tau.n(100)).real().round())^(1/3) 640320
""" CC = ComplexField() try: z = CC(z) except ValueError: raise ValueError("elliptic_j only defined for complex arguments.")
#### elliptic integrals
class EllipticE(BuiltinFunction): r""" Return the incomplete elliptic integral of the second kind:
.. MATH::
E(\varphi\,|\,m)=\int_0^\varphi \sqrt{1 - m\sin(x)^2}\, dx.
EXAMPLES::
sage: z = var("z") sage: elliptic_e(z, 1) elliptic_e(z, 1) sage: # this is still wrong: must be abs(sin(z)) + 2*round(z/pi) sage: elliptic_e(z, 1).simplify() 2*round(z/pi) + sin(z) sage: elliptic_e(z, 0) z sage: elliptic_e(0.5, 0.1) # abs tol 2e-15 0.498011394498832 sage: elliptic_e(1/2, 1/10).n(200) 0.4980113944988315331154610406...
.. SEEALSO::
- Taking `\varphi = \pi/2` gives :func:`elliptic_ec()<sage.functions.special.EllipticEC>`.
- Taking `\varphi = \operatorname{arc\,sin}(\operatorname{sn}(u,m))` gives :func:`elliptic_eu()<sage.functions.special.EllipticEU>`.
REFERENCES:
- :wikipedia:`Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind`
- :wikipedia:`Jacobi_elliptic_functions` """ def __init__(self): """ TESTS::
sage: loads(dumps(elliptic_e)) elliptic_e sage: elliptic_e(x, x)._sympy_() elliptic_e(x, x) """ # Maple conversion left out since it uses # k instead of m as the second argument conversions=dict(mathematica='EllipticE', maxima='elliptic_e', sympy='elliptic_e', ))
def _eval_(self, z, m): """ EXAMPLES::
sage: z = var("z") sage: elliptic_e(0, x) 0 sage: elliptic_e(pi/2, x) elliptic_ec(x) sage: elliptic_e(z, 0) z sage: elliptic_e(z, 1) elliptic_e(z, 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(+Infinity, x^2*e) """
def _evalf_(self, z, m, parent=None, algorithm=None): """ EXAMPLES::
sage: elliptic_e(0.5, 0.1) 0.498011394498832 sage: elliptic_e(1/2, 1/10).n(200) 0.4980113944988315331154610406... sage: elliptic_e(I, I).n() -0.189847437084712 + 1.03209769372160*I
TESTS:
This gave an error in Maxima (:trac:`15046`)::
sage: elliptic_e(2.5, 2.5) 0.535647771608740 + 1.63996015168665*I """
def _derivative_(self, z, m, diff_param): """ EXAMPLES::
sage: x,z = var('x,z') sage: elliptic_e(z, x).diff(z, 1) sqrt(-x*sin(z)^2 + 1) sage: elliptic_e(z, x).diff(x, 1) 1/2*(elliptic_e(z, x) - elliptic_f(z, x))/x """
def _print_latex_(self, z, m): """ EXAMPLES::
sage: latex(elliptic_e(pi, x)) E(\pi\,|\,x) """
elliptic_e = EllipticE()
class EllipticEC(BuiltinFunction): """ Return the complete elliptic integral of the second kind:
.. MATH::
E(m)=\int_0^{\pi/2} \sqrt{1 - m\sin(x)^2}\, dx.
EXAMPLES::
sage: elliptic_ec(0.1) 1.53075763689776 sage: elliptic_ec(x).diff() 1/2*(elliptic_ec(x) - elliptic_kc(x))/x
.. SEEALSO::
- :func:`elliptic_e()<sage.functions.special.EllipticE>`.
REFERENCES:
- :wikipedia:`Elliptic_integral#Complete_elliptic_integral_of_the_second_kind` """ def __init__(self): """ EXAMPLES::
sage: loads(dumps(elliptic_ec)) elliptic_ec sage: elliptic_ec(x)._sympy_() elliptic_e(x) """ conversions=dict(mathematica='EllipticE', maxima='elliptic_ec', sympy='elliptic_e', fricas='ellipticE'))
def _eval_(self, x): """ EXAMPLES::
sage: elliptic_ec(0) 1/2*pi sage: elliptic_ec(1) 1 sage: elliptic_ec(x) elliptic_ec(x) """
def _evalf_(self, x, parent=None, algorithm=None): """ EXAMPLES::
sage: elliptic_ec(sqrt(2)/2).n() 1.23742252487318 sage: elliptic_ec(sqrt(2)/2).n(200) 1.237422524873181672854746084083... sage: elliptic_ec(I).n() 1.63241178144043 - 0.369219492375499*I """
def _derivative_(self, x, diff_param): """ EXAMPLES::
sage: elliptic_ec(x).diff() 1/2*(elliptic_ec(x) - elliptic_kc(x))/x """
elliptic_ec = EllipticEC()
class EllipticEU(BuiltinFunction): r""" Return Jacobi's form of the incomplete elliptic integral of the second kind:
.. MATH::
E(u,m)= \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)`.
Also, ``elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)``.
EXAMPLES::
sage: elliptic_eu (0.5, 0.1) 0.496054551286597
.. SEEALSO::
- :func:`elliptic_e()<sage.functions.special.EllipticE>`.
REFERENCES:
- :wikipedia:`Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind`
- :wikipedia:`Jacobi_elliptic_functions` """ def __init__(self): r""" EXAMPLES::
sage: loads(dumps(elliptic_eu)) elliptic_eu """ conversions=dict(maxima='elliptic_eu'))
def _eval_(self, u, m): """ EXAMPLES::
sage: elliptic_eu(1,1) elliptic_eu(1, 1) """
def _evalf_(self, u, m, parent=None, algorithm=None): """ EXAMPLES::
sage: elliptic_eu(1,1).n() 0.761594155955765 sage: elliptic_eu(1,1).n(200) 0.7615941559557648881194582... """
def _derivative_(self, u, m, diff_param): """ EXAMPLES::
sage: x,m = var('x,m') sage: elliptic_eu(x,m).diff(x) sqrt(-m*jacobi_sn(x, m)^2 + 1)*jacobi_dn(x, m) sage: elliptic_eu(x,m).diff(m) 1/2*(elliptic_eu(x, m) - elliptic_f(jacobi_am(x, m), m))/m - 1/2*(m*jacobi_cn(x, m)*jacobi_sn(x, m) - (m - 1)*x - elliptic_eu(x, m)*jacobi_dn(x, m))*sqrt(-m*jacobi_sn(x, m)^2 + 1)/((m - 1)*m) """ Integer(1)) * jacobi('dn', u, m)) (elliptic_eu(u, m) - elliptic_f(jacobi_am(u, m), m)) / m - Integer(1) / Integer(2) * sqrt(-m * jacobi('sn', u, m) ** Integer(2) + Integer(1)) * (m * jacobi('sn', u, m) * jacobi('cn', u, m) - (m - Integer(1)) * u - elliptic_eu(u, m) * jacobi('dn', u, m)) / ((m - Integer(1)) * m))
def _print_latex_(self, u, m): """ EXAMPLES::
sage: latex(elliptic_eu(1,x)) E(1;x) """
def elliptic_eu_f(u, m): r""" Internal function for numeric evaluation of ``elliptic_eu``, defined as `E\left(\operatorname{am}(u, m)|m\right)`, where `E` is the incomplete elliptic integral of the second kind and `\operatorname{am}` is the Jacobi amplitude function.
EXAMPLES::
sage: from sage.functions.special import elliptic_eu_f sage: elliptic_eu_f(0.5, 0.1) mpf('0.49605455128659691') """
finally:
elliptic_eu = EllipticEU()
class EllipticF(BuiltinFunction): r""" Return the incomplete elliptic integral of the first kind.
.. MATH::
F(\varphi\,|\,m)=\int_0^\varphi \frac{dx}{\sqrt{1 - m\sin(x)^2}},
Taking `\varphi = \pi/2` gives :func:`elliptic_kc()<sage.functions.special.EllipticKC>`.
EXAMPLES::
sage: z = var("z") sage: elliptic_f (z, 0) z sage: elliptic_f (z, 1).simplify() log(tan(1/4*pi + 1/2*z)) sage: elliptic_f (0.2, 0.1) 0.200132506747543
.. SEEALSO::
- :func:`elliptic_e()<sage.functions.special.EllipticE>`.
REFERENCES:
- :wikipedia:`Elliptic_integral#Incomplete_elliptic_integral_of_the_first_kind` """ def __init__(self): """ EXAMPLES::
sage: loads(dumps(elliptic_f)) elliptic_f sage: elliptic_f(x, 2)._sympy_() elliptic_f(x, 2) """ conversions=dict(mathematica='EllipticF', maxima='elliptic_f', sympy='elliptic_f'))
def _eval_(self, z, m): """ EXAMPLES::
sage: elliptic_f(x,1) elliptic_f(x, 1) sage: elliptic_f(x,0) x sage: elliptic_f(0,1) 0 sage: elliptic_f(pi/2,x) elliptic_kc(x) """
def _evalf_(self, z, m, parent=None, algorithm=None): """ EXAMPLES::
sage: elliptic_f(1,1).n() 1.22619117088352 sage: elliptic_f(1,1).n(200) 1.22619117088351707081306096... sage: elliptic_f(I,I).n() 0.149965060031782 + 0.925097284105771*I """
def _derivative_(self, z, m, diff_param): """ EXAMPLES::
sage: x,m = var('x,m') sage: elliptic_f(x,m).diff(x) 1/sqrt(-m*sin(x)^2 + 1) sage: elliptic_f(x,m).diff(m) -1/2*elliptic_f(x, m)/m + 1/4*sin(2*x)/(sqrt(-m*sin(x)^2 + 1)*(m - 1)) - 1/2*elliptic_e(x, m)/((m - 1)*m) """ elliptic_f(z, m) / (Integer(2) * m) - (sin(Integer(2) * z) / (Integer(4) * (Integer(1) - m) * sqrt(Integer(1) - m * sin(z) ** Integer(2)))))
def _print_latex_(self, z, m): """ EXAMPLES::
sage: latex(elliptic_f(x,pi)) F(x\,|\,\pi) """
elliptic_f = EllipticF()
class EllipticKC(BuiltinFunction): r""" Return the complete elliptic integral of the first kind:
.. MATH::
K(m)=\int_0^{\pi/2} \frac{dx}{\sqrt{1 - m\sin(x)^2}}.
EXAMPLES::
sage: elliptic_kc(0.5) 1.85407467730137
.. SEEALSO::
- :func:`elliptic_f()<sage.functions.special.EllipticF>`.
- :func:`elliptic_ec()<sage.functions.special.EllipticEC>`.
REFERENCES:
- :wikipedia:`Elliptic_integral#Complete_elliptic_integral_of_the_first_kind`
- :wikipedia:`Elliptic_integral#Incomplete_elliptic_integral_of_the_first_kind` """ def __init__(self): """ EXAMPLES::
sage: loads(dumps(elliptic_kc)) elliptic_kc sage: elliptic_kc(x)._sympy_() elliptic_k(x) """ conversions=dict(mathematica='EllipticK', maxima='elliptic_kc', sympy='elliptic_k', fricas='ellipticK'))
def _eval_(self, z): """ EXAMPLES::
sage: elliptic_kc(0) 1/2*pi sage: elliptic_kc(1/2) elliptic_kc(1/2)
TESTS:
Check if complex numbers in the arguments are converted to maxima correctly (see :trac:`7557`)::
sage: t = jacobi_sn(1.2+2*I*elliptic_kc(1-.5),.5) sage: maxima(t) # abs tol 1e-13 0.88771548861928029 - 1.7301614091485560e-15*%i sage: t.n() # abs tol 1e-13 0.887715488619280 - 1.73016140914856e-15*I """ else:
def _evalf_(self, z, parent=None, algorithm=None): """ EXAMPLES::
sage: elliptic_kc(1/2).n() 1.85407467730137 sage: elliptic_kc(1/2).n(200) 1.85407467730137191843385034... sage: elliptic_kc(I).n() 1.42127228104504 + 0.295380284214777*I """
def _derivative_(self, z, diff_param): """ EXAMPLES::
sage: elliptic_kc(x).diff(x) -1/2*((x - 1)*elliptic_kc(x) + elliptic_ec(x))/((x - 1)*x) """ (Integer(2) * (Integer(1) - z) * z))
elliptic_kc = EllipticKC()
class EllipticPi(BuiltinFunction): r""" Return the incomplete elliptic integral of the third kind:
.. MATH::
\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) 0.200665068220979 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)
REFERENCES:
- :wikipedia:`Elliptic_integral#Incomplete_elliptic_integral_of_the_third_kind` """ def __init__(self): """ EXAMPLES::
sage: loads(dumps(elliptic_pi)) elliptic_pi sage: elliptic_pi(x, pi/4, 1)._sympy_() elliptic_pi(x, pi/4, 1) """ conversions=dict(mathematica='EllipticPi', maxima='EllipticPi', sympy='elliptic_pi'))
def _eval_(self, n, z, m): """ EXAMPLES::
sage: elliptic_pi(x,x,pi) elliptic_pi(x, x, pi) sage: elliptic_pi(0,x,pi) elliptic_f(x, pi) """
def _evalf_(self, n, z, m, parent=None, algorithm=None): """ EXAMPLES::
sage: elliptic_pi(pi,1/2,1).n() 0.795062820631931 sage: elliptic_pi(pi,1/2,1).n(200) 0.79506282063193125292514098445... sage: elliptic_pi(pi,1,1).n() 0.0991592574231369 - 1.30004368185937*I sage: elliptic_pi(pi,I,I).n() 0.0542471560940594 + 0.552096453413081*I """
def _derivative_(self, n, z, m, diff_param): """ EXAMPLES::
sage: n,z,m = var('n,z,m') sage: elliptic_pi(n,z,m).diff(n) 1/4*(sqrt(-m*sin(z)^2 + 1)*n*sin(2*z)/(n*sin(z)^2 - 1) + 2*(m - n)*elliptic_f(z, m)/n + 2*(n^2 - m)*elliptic_pi(n, z, m)/n + 2*elliptic_e(z, m))/((m - n)*(n - 1)) sage: elliptic_pi(n,z,m).diff(z) -1/(sqrt(-m*sin(z)^2 + 1)*(n*sin(z)^2 - 1)) sage: elliptic_pi(n,z,m).diff(m) 1/4*(m*sin(2*z)/(sqrt(-m*sin(z)^2 + 1)*(m - 1)) - 2*elliptic_e(z, m)/(m - 1) - 2*elliptic_pi(n, z, m))/(m - n) """ (elliptic_e(z, m) + ((m - n) / n) * elliptic_f(z, m) + ((n ** Integer(2) - m) / n) * elliptic_pi(n, z, m) - (n * sqrt(Integer(1) - m * sin(z) ** Integer(2)) * sin(Integer(2) * z)) / (Integer(2) * (Integer(1) - n * sin(z) ** Integer(2))))) (sqrt(Integer(1) - m * sin(z) ** Integer(Integer(2))) * (Integer(1) - n * sin(z) ** Integer(2)))) (elliptic_e(z, m) / (m - Integer(1)) + elliptic_pi(n, z, m) - (m * sin(Integer(2) * z)) / (Integer(2) * (m - Integer(1)) * sqrt(Integer(1) - m * sin(z) ** Integer(2)))))
def _print_latex_(self, n, z, m): """ EXAMPLES::
sage: latex(elliptic_pi(x,pi,0)) \Pi(x,\pi,0) """
elliptic_pi = EllipticPi()
def error_fcn(x): """ Deprecated in :trac:`21819`. Please use ``erfc()``.
EXAMPLES::
sage: error_fcn(x) doctest:warning ... DeprecationWarning: error_fcn() is deprecated. Please use erfc() See http://trac.sagemath.org/21819 for details. erfc(x) """
|