Source code for diofant.functions.special.polynomials

"""
This module mainly implements special orthogonal polynomials.

See also functions.combinatorial.numbers which contains some
combinatorial polynomials.

"""

from ...core import Dummy, Function, Integer, Rational, oo, pi, zoo
from ...core.function import ArgumentIndexError
from ...polys.orthopolys import (chebyshevt_poly, chebyshevu_poly,
                                 gegenbauer_poly, hermite_poly, jacobi_poly,
                                 laguerre_poly, legendre_poly)
from ..combinatorial.factorials import RisingFactorial, binomial, factorial
from ..elementary.complexes import re
from ..elementary.exponential import exp
from ..elementary.miscellaneous import sqrt
from ..elementary.trigonometric import cos
from .gamma_functions import gamma
from .hyper import hyper


_x = Dummy('dummy_for_special_polynomials')


class OrthogonalPolynomial(Function):
    """Base class for orthogonal polynomials."""

    @classmethod
    def _eval_at_order(cls, n, x):
        if n.is_integer and n >= 0:
            return cls._ortho_poly(int(n), _x).subs({_x: x})

    def _eval_conjugate(self):
        return self.func(self.args[0], self.args[1].conjugate())

############################################################################
# Jacobi polynomials
#


[docs]class jacobi(OrthogonalPolynomial): r""" Jacobi polynomial `P_n^{\left(\alpha, \beta\right)}(x)` jacobi(n, alpha, beta, x) gives the nth Jacobi polynomial in x, `P_n^{\left(\alpha, \beta\right)}(x)`. The Jacobi polynomials are orthogonal on `[-1, 1]` with respect to the weight `\left(1-x\right)^\alpha \left(1+x\right)^\beta`. Examples ======== >>> jacobi(0, a, b, x) 1 >>> jacobi(1, a, b, x) a/2 - b/2 + x*(a/2 + b/2 + 1) >>> jacobi(n, a, b, x) jacobi(n, a, b, x) >>> jacobi(n, a, a, x) RisingFactorial(a + 1, n)*gegenbauer(n, a + 1/2, x)/RisingFactorial(2*a + 1, n) >>> jacobi(n, 0, 0, x) legendre(n, x) >>> jacobi(n, Rational(1, 2), Rational(1, 2), x) RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1) >>> jacobi(n, -Rational(1, 2), -Rational(1, 2), x) RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n) >>> jacobi(n, a, b, -x) (-1)**n*jacobi(n, b, a, x) >>> jacobi(n, a, b, 0) 2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1)) >>> jacobi(n, a, b, 1) RisingFactorial(a + 1, n)/factorial(n) >>> conjugate(jacobi(n, a, b, x)) jacobi(n, conjugate(a), conjugate(b), conjugate(x)) >>> diff(jacobi(n, a, b, x), x) (a + b + n + 1)*jacobi(n - 1, a + 1, b + 1, x)/2 See Also ======== gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly, diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * https://en.wikipedia.org/wiki/Jacobi_polynomials * http://mathworld.wolfram.com/JacobiPolynomial.html * http://functions.wolfram.com/Polynomials/JacobiP/ """ @classmethod def eval(cls, n, a, b, x): # Simplify to other polynomials # P^{a, a}_n(x) if a == b: if a == -Rational(1, 2): return RisingFactorial(Rational(1, 2), n) / factorial(n) * chebyshevt(n, x) elif a == 0: return legendre(n, x) elif a == Rational(1, 2): return RisingFactorial(Rational(3, 2), n) / factorial(n + 1) * chebyshevu(n, x) else: return RisingFactorial(a + 1, n) / RisingFactorial(2*a + 1, n) * gegenbauer(n, a + Rational(1, 2), x) elif b == -a: # P^{a, -a}_n(x) return gamma(n + a + 1) / gamma(n + 1) * (1 + x)**(a/2) / (1 - x)**(a/2) * assoc_legendre(n, -a, x) if not n.is_Number: # Symbolic result P^{a,b}_n(x) # P^{a,b}_n(-x) ---> (-1)**n * P^{b,a}_n(-x) if x.could_extract_minus_sign(): return (-1)**n * jacobi(n, b, a, -x) # We can evaluate for some special values of x if x == 0: return (2**(-n) * gamma(a + n + 1) / (gamma(a + 1) * factorial(n)) * hyper([-b - n, -n], [a + 1], -1)) if x == 1: return RisingFactorial(a + 1, n) / factorial(n) elif x == oo: if n.is_positive: # Make sure a+b+2*n \notin Z if (a + b + 2*n).is_integer: raise ValueError("a + b + 2*n should not be an integer.") return RisingFactorial(a + b + n + 1, n) * oo else: # n is a given fixed integer, evaluate into polynomial return jacobi_poly(n, a, b, x) def fdiff(self, argindex=4): from ...concrete import Sum if argindex == 1: # Diff wrt n raise ArgumentIndexError(self, argindex) elif argindex == 2: # Diff wrt a n, a, b, x = self.args k = Dummy("k") f1 = 1 / (a + b + n + k + 1) f2 = ((a + b + 2*k + 1) * RisingFactorial(b + k + 1, n - k) / ((n - k) * RisingFactorial(a + b + k + 1, n - k))) return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1)) elif argindex == 3: # Diff wrt b n, a, b, x = self.args k = Dummy("k") f1 = 1 / (a + b + n + k + 1) f2 = (-1)**(n - k) * ((a + b + 2*k + 1) * RisingFactorial(a + k + 1, n - k) / ((n - k) * RisingFactorial(a + b + k + 1, n - k))) return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1)) elif argindex == 4: # Diff wrt x n, a, b, x = self.args return (a + b + n + 1) * jacobi(n - 1, a + 1, b + 1, x) / 2 else: raise ArgumentIndexError(self, argindex) def _eval_conjugate(self): n, a, b, x = self.args return self.func(n, a.conjugate(), b.conjugate(), x.conjugate())
[docs]def jacobi_normalized(n, a, b, x): r""" Jacobi polynomial `P_n^{\left(\alpha, \beta\right)}(x)` jacobi_normalized(n, alpha, beta, x) gives the nth Jacobi polynomial in x, `P_n^{\left(\alpha, \beta\right)}(x)`. The Jacobi polynomials are orthogonal on `[-1, 1]` with respect to the weight `\left(1-x\right)^\alpha \left(1+x\right)^\beta`. This functions returns the polynomials normilzed: .. math:: \int_{-1}^{1} P_m^{\left(\alpha, \beta\right)}(x) P_n^{\left(\alpha, \beta\right)}(x) (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x = \delta_{m,n} Examples ======== >>> jacobi_normalized(n, a, b, x) jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))) See Also ======== gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly, diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * https://en.wikipedia.org/wiki/Jacobi_polynomials * http://mathworld.wolfram.com/JacobiPolynomial.html * http://functions.wolfram.com/Polynomials/JacobiP/ """ nfactor = (Integer(2)**(a + b + 1) * (gamma(n + a + 1) * gamma(n + b + 1)) / (2*n + a + b + 1) / (factorial(n) * gamma(n + a + b + 1))) return jacobi(n, a, b, x) / sqrt(nfactor)
############################################################################ # Gegenbauer polynomials #
[docs]class gegenbauer(OrthogonalPolynomial): r""" Gegenbauer polynomial `C_n^{\left(\alpha\right)}(x)` gegenbauer(n, alpha, x) gives the nth Gegenbauer polynomial in x, `C_n^{\left(\alpha\right)}(x)`. The Gegenbauer polynomials are orthogonal on `[-1, 1]` with respect to the weight `\left(1-x^2\right)^{\alpha-\frac{1}{2}}`. Examples ======== >>> gegenbauer(0, a, x) 1 >>> gegenbauer(1, a, x) 2*a*x >>> gegenbauer(2, a, x) -a + x**2*(2*a**2 + 2*a) >>> gegenbauer(3, a, x) x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a) >>> gegenbauer(n, a, x) gegenbauer(n, a, x) >>> gegenbauer(n, a, -x) (-1)**n*gegenbauer(n, a, x) >>> gegenbauer(n, a, 0) 2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(-n/2 + 1/2)*gamma(n + 1)) >>> gegenbauer(n, a, 1) gamma(2*a + n)/(gamma(2*a)*gamma(n + 1)) >>> conjugate(gegenbauer(n, a, x)) gegenbauer(n, conjugate(a), conjugate(x)) >>> diff(gegenbauer(n, a, x), x) 2*a*gegenbauer(n - 1, a + 1, x) See Also ======== jacobi, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * https://en.wikipedia.org/wiki/Gegenbauer_polynomials * http://mathworld.wolfram.com/GegenbauerPolynomial.html * http://functions.wolfram.com/Polynomials/GegenbauerC3/ """ @classmethod def eval(cls, n, a, x): # For negative n the polynomials vanish # See http://functions.wolfram.com/Polynomials/GegenbauerC3/03/01/03/0012/ if n.is_negative: return Integer(0) # Some special values for fixed a if a == Rational(1, 2): return legendre(n, x) elif a == 1: return chebyshevu(n, x) elif a == -1: return Integer(0) if not n.is_Number: # Handle this before the general sign extraction rule if x == -1: if (re(a) - Rational(1, 2)).is_positive: return zoo else: return # Symbolic result C^a_n(x) # C^a_n(-x) ---> (-1)**n * C^a_n(x) if x.could_extract_minus_sign(): return (-1)**n * gegenbauer(n, a, -x) # We can evaluate for some special values of x if x == 0: return (2**n * sqrt(pi) * gamma(a + n/2) / (gamma((1 - n)/2) * gamma(n + 1) * gamma(a)) ) if x == 1: return gamma(2*a + n) / (gamma(2*a) * gamma(n + 1)) elif x == oo: if n.is_positive: return RisingFactorial(a, n) * oo else: # n is a given fixed integer, evaluate into polynomial return gegenbauer_poly(n, a, x) def fdiff(self, argindex=3): from ...concrete import Sum if argindex == 1: # Diff wrt n raise ArgumentIndexError(self, argindex) elif argindex == 2: # Diff wrt a n, a, x = self.args k = Dummy("k") factor1 = 2 * (1 + (-1)**(n - k)) * (k + a) / ((k + n + 2*a) * (n - k)) factor2 = 2*(k + 1) / ((k + 2*a) * (2*k + 2*a + 1)) + \ 2 / (k + n + 2*a) kern = factor1*gegenbauer(k, a, x) + factor2*gegenbauer(n, a, x) return Sum(kern, (k, 0, n - 1)) elif argindex == 3: # Diff wrt x n, a, x = self.args return 2*a*gegenbauer(n - 1, a + 1, x) else: raise ArgumentIndexError(self, argindex) def _eval_conjugate(self): n, a, x = self.args return self.func(n, a.conjugate(), x.conjugate())
############################################################################ # Chebyshev polynomials of first and second kind #
[docs]class chebyshevt(OrthogonalPolynomial): r""" Chebyshev polynomial of the first kind, `T_n(x)` chebyshevt(n, x) gives the nth Chebyshev polynomial (of the first kind) in x, `T_n(x)`. The Chebyshev polynomials of the first kind are orthogonal on `[-1, 1]` with respect to the weight `\frac{1}{\sqrt{1-x^2}}`. Examples ======== >>> chebyshevt(0, x) 1 >>> chebyshevt(1, x) x >>> chebyshevt(2, x) 2*x**2 - 1 >>> chebyshevt(n, x) chebyshevt(n, x) >>> chebyshevt(n, -x) (-1)**n*chebyshevt(n, x) >>> chebyshevt(-n, x) chebyshevt(n, x) >>> chebyshevt(n, 0) cos(pi*n/2) >>> chebyshevt(n, -1) (-1)**n >>> diff(chebyshevt(n, x), x) n*chebyshevu(n - 1, x) See Also ======== jacobi, gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * https://en.wikipedia.org/wiki/Chebyshev_polynomial * http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html * http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html * http://functions.wolfram.com/Polynomials/ChebyshevT/ * http://functions.wolfram.com/Polynomials/ChebyshevU/ """ _ortho_poly = staticmethod(chebyshevt_poly) @classmethod def eval(cls, n, x): if not n.is_Number: # Symbolic result T_n(x) # T_n(-x) ---> (-1)**n * T_n(x) if x.could_extract_minus_sign(): return (-1)**n * chebyshevt(n, -x) # T_{-n}(x) ---> T_n(x) if n.could_extract_minus_sign(): return chebyshevt(-n, x) # We can evaluate for some special values of x if x == 0: return cos(pi*n/2) if x == 1: return Integer(1) elif x == oo: return oo else: # n is a given fixed integer, evaluate into polynomial if n.is_negative: # T_{-n}(x) == T_n(x) return cls._eval_at_order(-n, x) else: return cls._eval_at_order(n, x) def fdiff(self, argindex=2): if argindex == 2: # Diff wrt x n, x = self.args return n * chebyshevu(n - 1, x) else: # wrt n raise ArgumentIndexError(self, argindex)
[docs]class chebyshevu(OrthogonalPolynomial): r""" Chebyshev polynomial of the second kind, `U_n(x)` chebyshevu(n, x) gives the nth Chebyshev polynomial of the second kind in x, `U_n(x)`. The Chebyshev polynomials of the second kind are orthogonal on `[-1, 1]` with respect to the weight `\sqrt{1-x^2}`. Examples ======== >>> chebyshevu(0, x) 1 >>> chebyshevu(1, x) 2*x >>> chebyshevu(2, x) 4*x**2 - 1 >>> chebyshevu(n, x) chebyshevu(n, x) >>> chebyshevu(n, -x) (-1)**n*chebyshevu(n, x) >>> chebyshevu(-n, x) -chebyshevu(n - 2, x) >>> chebyshevu(n, 0) cos(pi*n/2) >>> chebyshevu(n, 1) n + 1 >>> diff(chebyshevu(n, x), x) (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * https://en.wikipedia.org/wiki/Chebyshev_polynomial * http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html * http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html * http://functions.wolfram.com/Polynomials/ChebyshevT/ * http://functions.wolfram.com/Polynomials/ChebyshevU/ """ _ortho_poly = staticmethod(chebyshevu_poly) @classmethod def eval(cls, n, x): if not n.is_Number: # Symbolic result U_n(x) # U_n(-x) ---> (-1)**n * U_n(x) if x.could_extract_minus_sign(): return (-1)**n * chebyshevu(n, -x) # U_{-n}(x) ---> -U_{n-2}(x) if n.could_extract_minus_sign(): return -chebyshevu(-n - 2, x) # We can evaluate for some special values of x if x == 0: return cos(pi*n/2) if x == 1: return 1 + n elif x == oo: return oo else: # n is a given fixed integer, evaluate into polynomial if n.is_negative: # U_{-n}(x) ---> -U_{n-2}(x) if n == -1: return Integer(0) else: return -cls._eval_at_order(-n - 2, x) else: return cls._eval_at_order(n, x) def fdiff(self, argindex=2): if argindex == 2: # Diff wrt x n, x = self.args return ((n + 1) * chebyshevt(n + 1, x) - x * chebyshevu(n, x)) / (x**2 - 1) else: # wrt n raise ArgumentIndexError(self, argindex)
[docs]class chebyshevt_root(Function): r""" chebyshev_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the first kind; that is, if 0 <= k < n, chebyshevt(n, chebyshevt_root(n, k)) == 0. Examples ======== >>> chebyshevt_root(3, 2) -sqrt(3)/2 >>> chebyshevt(3, chebyshevt_root(3, 2)) 0 See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly """ @classmethod def eval(cls, n, k): if not ((0 <= k) and (k < n)): raise ValueError("must have 0 <= k < n, " "got k = %s and n = %s" % (k, n)) return cos(pi*(2*k + 1)/(2*n))
[docs]class chebyshevu_root(Function): r""" chebyshevu_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the second kind; that is, if 0 <= k < n, chebyshevu(n, chebyshevu_root(n, k)) == 0. Examples ======== >>> chebyshevu_root(3, 2) -sqrt(2)/2 >>> chebyshevu(3, chebyshevu_root(3, 2)) 0 See Also ======== chebyshevt, chebyshevt_root, chebyshevu, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly """ @classmethod def eval(cls, n, k): if not ((0 <= k) and (k < n)): raise ValueError("must have 0 <= k < n, " "got k = %s and n = %s" % (k, n)) return cos(pi*(k + 1)/(n + 1))
############################################################################ # Legendre polynomials and Associated Legendre polynomials #
[docs]class legendre(OrthogonalPolynomial): r""" legendre(n, x) gives the nth Legendre polynomial of x, `P_n(x)` The Legendre polynomials are orthogonal on [-1, 1] with respect to the constant weight 1. They satisfy `P_n(1) = 1` for all n; further, `P_n` is odd for odd n and even for even n. Examples ======== >>> legendre(0, x) 1 >>> legendre(1, x) x >>> legendre(2, x) 3*x**2/2 - 1/2 >>> legendre(n, x) legendre(n, x) >>> diff(legendre(n, x), x) n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, assoc_legendre, hermite, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * https://en.wikipedia.org/wiki/Legendre_polynomial * http://mathworld.wolfram.com/LegendrePolynomial.html * http://functions.wolfram.com/Polynomials/LegendreP/ * http://functions.wolfram.com/Polynomials/LegendreP2/ """ _ortho_poly = staticmethod(legendre_poly) @classmethod def eval(cls, n, x): if not n.is_Number: # Symbolic result L_n(x) # L_n(-x) ---> (-1)**n * L_n(x) if x.could_extract_minus_sign(): return (-1)**n * legendre(n, -x) # L_{-n}(x) ---> L_{n-1}(x) if n.could_extract_minus_sign(): return legendre(-n - Integer(1), x) # We can evaluate for some special values of x if x == 0: return sqrt(pi)/(gamma((1 - n)/2)*gamma(Integer(1) + n/2)) elif x == 1: return Integer(1) elif x == oo: return oo else: # n is a given fixed integer, evaluate into polynomial if n.is_negative: raise ValueError( "The index n must be nonnegative integer (got %r)" % n) else: return cls._eval_at_order(n, x) def fdiff(self, argindex=2): if argindex == 2: # Diff wrt x # Find better formula, this is unsuitable for x = 1 n, x = self.args return n/(x**2 - 1)*(x*legendre(n, x) - legendre(n - 1, x)) else: # wrt n raise ArgumentIndexError(self, argindex)
[docs]class assoc_legendre(Function): r""" assoc_legendre(n,m, x) gives `P_n^m(x)`, where n and m are the degree and order or an expression which is related to the nth order Legendre polynomial, `P_n(x)` in the following manner: .. math:: P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}} \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m} Associated Legendre polynomial are orthogonal on [-1, 1] with: - weight = 1 for the same m, and different n. - weight = 1/(1-x**2) for the same n, and different m. Examples ======== >>> assoc_legendre(0, 0, x) 1 >>> assoc_legendre(1, 0, x) x >>> assoc_legendre(1, 1, x) -sqrt(-x**2 + 1) >>> assoc_legendre(n, m, x) assoc_legendre(n, m, x) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, hermite, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * https://en.wikipedia.org/wiki/Associated_Legendre_polynomials * http://mathworld.wolfram.com/LegendrePolynomial.html * http://functions.wolfram.com/Polynomials/LegendreP/ * http://functions.wolfram.com/Polynomials/LegendreP2/ """ @classmethod def _eval_at_order(cls, n, m): P = legendre_poly(n, _x, polys=True).diff((_x, m)) return (-1)**m * (1 - _x**2)**Rational(m, 2) * P.as_expr() @classmethod def eval(cls, n, m, x): if m.could_extract_minus_sign(): # P^{-m}_n ---> F * P^m_n return (-1)**(-m) * (factorial(m + n)/factorial(n - m)) * assoc_legendre(n, -m, x) if m == 0: # P^0_n ---> L_n return legendre(n, x) if x == 0: return 2**m*sqrt(pi) / (gamma((1 - m - n)/2)*gamma(1 - (m - n)/2)) if n.is_Number and m.is_Number and n.is_integer and m.is_integer: if n.is_negative: raise ValueError("%s : 1st index must be nonnegative integer (got %r)" % (cls, n)) if abs(m) > n: raise ValueError("%s : abs('2nd index') must be <= '1st index' (got %r, %r)" % (cls, n, m)) return cls._eval_at_order(int(n), abs(int(m))).subs({_x: x}) def fdiff(self, argindex=3): if argindex == 3: # Diff wrt x # Find better formula, this is unsuitable for x = 1 n, m, x = self.args return 1/(x**2 - 1)*(x*n*assoc_legendre(n, m, x) - (m + n)*assoc_legendre(n - 1, m, x)) else: # wrt n, m raise ArgumentIndexError(self, argindex) def _eval_conjugate(self): n, m, x = self.args return self.func(n, m.conjugate(), x.conjugate())
############################################################################ # Hermite polynomials #
[docs]class hermite(OrthogonalPolynomial): r""" hermite(n, x) gives the nth Hermite polynomial in x, `H_n(x)` The Hermite polynomials are orthogonal on `(-\infty, \infty)` with respect to the weight `\exp\left(-x^2\right)`. Examples ======== >>> hermite(0, x) 1 >>> hermite(1, x) 2*x >>> hermite(2, x) 4*x**2 - 2 >>> hermite(n, x) hermite(n, x) >>> diff(hermite(n, x), x) 2*n*hermite(n - 1, x) >>> hermite(n, -x) (-1)**n*hermite(n, x) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, laguerre, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * https://en.wikipedia.org/wiki/Hermite_polynomial * http://mathworld.wolfram.com/HermitePolynomial.html * http://functions.wolfram.com/Polynomials/HermiteH/ """ _ortho_poly = staticmethod(hermite_poly) @classmethod def eval(cls, n, x): if not n.is_Number: # Symbolic result H_n(x) # H_n(-x) ---> (-1)**n * H_n(x) if x.could_extract_minus_sign(): return (-1)**n * hermite(n, -x) # We can evaluate for some special values of x if x == 0: return 2**n * sqrt(pi) / gamma((1 - n)/2) elif x == oo: return oo else: # n is a given fixed integer, evaluate into polynomial if n.is_negative: raise ValueError( "The index n must be nonnegative integer (got %r)" % n) else: return cls._eval_at_order(n, x) def fdiff(self, argindex=2): if argindex == 2: # Diff wrt x n, x = self.args return 2*n*hermite(n - 1, x) else: # wrt n raise ArgumentIndexError(self, argindex)
############################################################################ # Laguerre polynomials #
[docs]class laguerre(OrthogonalPolynomial): r""" Returns the nth Laguerre polynomial in x, `L_n(x)`. Parameters ========== n : int Degree of Laguerre polynomial. Must be ``n >= 0``. Examples ======== >>> laguerre(0, x) 1 >>> laguerre(1, x) -x + 1 >>> laguerre(2, x) x**2/2 - 2*x + 1 >>> laguerre(3, x) -x**3/6 + 3*x**2/2 - 3*x + 1 >>> laguerre(n, x) laguerre(n, x) >>> diff(laguerre(n, x), x) -assoc_laguerre(n - 1, 1, x) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, assoc_laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * https://en.wikipedia.org/wiki/Laguerre_polynomial * http://mathworld.wolfram.com/LaguerrePolynomial.html * http://functions.wolfram.com/Polynomials/LaguerreL/ * http://functions.wolfram.com/Polynomials/LaguerreL3/ """ _ortho_poly = staticmethod(laguerre_poly) @classmethod def eval(cls, n, x): if not n.is_Number: # Symbolic result L_n(x) # L_{n}(-x) ---> exp(-x) * L_{-n-1}(x) # L_{-n}(x) ---> exp(x) * L_{n-1}(-x) if n.could_extract_minus_sign(): return exp(x) * laguerre(-n - 1, -x) # We can evaluate for some special values of x if x == 0: return Integer(1) elif x == -oo: return oo elif x == oo: return (-1)**n * oo else: # n is a given fixed integer, evaluate into polynomial if n.is_negative: raise ValueError( "The index n must be nonnegative integer (got %r)" % n) else: return cls._eval_at_order(n, x) def fdiff(self, argindex=2): if argindex == 2: # Diff wrt x n, x = self.args return -assoc_laguerre(n - 1, 1, x) else: # wrt n raise ArgumentIndexError(self, argindex)
[docs]class assoc_laguerre(OrthogonalPolynomial): r""" Returns the nth generalized Laguerre polynomial in x, `L_n(x)`. Parameters ========== n : int Degree of Laguerre polynomial. Must be ``n >= 0``. alpha : Expr Arbitrary expression. For ``alpha=0`` regular Laguerre polynomials will be generated. Examples ======== >>> assoc_laguerre(0, a, x) 1 >>> assoc_laguerre(1, a, x) a - x + 1 >>> assoc_laguerre(2, a, x) a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1 >>> assoc_laguerre(3, a, x) a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) + x*(-a**2/2 - 5*a/2 - 3) + 1 >>> assoc_laguerre(n, a, 0) binomial(a + n, a) >>> assoc_laguerre(n, a, x) assoc_laguerre(n, a, x) >>> assoc_laguerre(n, 0, x) laguerre(n, x) >>> diff(assoc_laguerre(n, a, x), x) -assoc_laguerre(n - 1, a + 1, x) >>> diff(assoc_laguerre(n, a, x), a) Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1)) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, diofant.polys.orthopolys.jacobi_poly diofant.polys.orthopolys.gegenbauer_poly diofant.polys.orthopolys.chebyshevt_poly diofant.polys.orthopolys.chebyshevu_poly diofant.polys.orthopolys.hermite_poly diofant.polys.orthopolys.legendre_poly diofant.polys.orthopolys.laguerre_poly References ========== * http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html * http://functions.wolfram.com/Polynomials/LaguerreL/ * http://functions.wolfram.com/Polynomials/LaguerreL3/ """ @classmethod def _eval_at_order(cls, n, x, alpha): if n.is_integer and n >= 0: return laguerre_poly(int(n), _x, alpha).subs({_x: x}) @classmethod def eval(cls, n, alpha, x): # L_{n}^{0}(x) ---> L_{n}(x) if alpha == 0: return laguerre(n, x) if not n.is_Number: # We can evaluate for some special values of x if x == 0: return binomial(n + alpha, alpha) elif x == oo and n.is_positive: return (-1)**n * oo elif x == -oo and n.is_positive: return oo else: # n is a given fixed integer, evaluate into polynomial if n.is_negative: raise ValueError( "The index n must be nonnegative integer (got %r)" % n) else: return cls._eval_at_order(n, x, alpha) def fdiff(self, argindex=3): from ...concrete import Sum if argindex == 2: # Diff wrt alpha n, alpha, x = self.args k = Dummy("k") return Sum(assoc_laguerre(k, alpha, x) / (n - alpha), (k, 0, n - 1)) elif argindex == 3: # Diff wrt x n, alpha, x = self.args return -assoc_laguerre(n - 1, alpha + 1, x) else: # wrt n raise ArgumentIndexError(self, argindex) def _eval_conjugate(self): n, alpha, x = self.args return self.func(n, alpha.conjugate(), x.conjugate())