1448 lines
46 KiB
Python
1448 lines
46 KiB
Python
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"""
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This module mainly implements special orthogonal polynomials.
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See also functions.combinatorial.numbers which contains some
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combinatorial polynomials.
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"""
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from sympy.core import Rational
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from sympy.core.function import Function, ArgumentIndexError
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from sympy.core.singleton import S
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from sympy.core.symbol import Dummy
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from sympy.functions.combinatorial.factorials import binomial, factorial, RisingFactorial
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from sympy.functions.elementary.complexes import re
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.integers import floor
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import cos, sec
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from sympy.functions.special.gamma_functions import gamma
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from sympy.functions.special.hyper import hyper
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from sympy.polys.orthopolys import (chebyshevt_poly, chebyshevu_poly,
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gegenbauer_poly, hermite_poly, hermite_prob_poly,
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jacobi_poly, laguerre_poly, legendre_poly)
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_x = Dummy('x')
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class OrthogonalPolynomial(Function):
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"""Base class for orthogonal polynomials.
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"""
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@classmethod
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def _eval_at_order(cls, n, x):
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if n.is_integer and n >= 0:
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return cls._ortho_poly(int(n), _x).subs(_x, x)
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def _eval_conjugate(self):
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return self.func(self.args[0], self.args[1].conjugate())
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#----------------------------------------------------------------------------
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# Jacobi polynomials
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#
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class jacobi(OrthogonalPolynomial):
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r"""
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Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.
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Explanation
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===========
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``jacobi(n, alpha, beta, x)`` gives the $n$th Jacobi polynomial
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in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$.
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The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
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to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.
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Examples
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========
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>>> from sympy import jacobi, S, conjugate, diff
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>>> from sympy.abc import a, b, n, x
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>>> jacobi(0, a, b, x)
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1
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>>> jacobi(1, a, b, x)
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a/2 - b/2 + x*(a/2 + b/2 + 1)
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>>> jacobi(2, a, b, x)
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a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2
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>>> jacobi(n, a, b, x)
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jacobi(n, a, b, x)
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>>> jacobi(n, a, a, x)
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RisingFactorial(a + 1, n)*gegenbauer(n,
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a + 1/2, x)/RisingFactorial(2*a + 1, n)
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>>> jacobi(n, 0, 0, x)
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legendre(n, x)
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>>> jacobi(n, S(1)/2, S(1)/2, x)
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RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)
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>>> jacobi(n, -S(1)/2, -S(1)/2, x)
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RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)
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>>> jacobi(n, a, b, -x)
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(-1)**n*jacobi(n, b, a, x)
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>>> jacobi(n, a, b, 0)
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gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(2**n*factorial(n)*gamma(a + 1))
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>>> jacobi(n, a, b, 1)
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RisingFactorial(a + 1, n)/factorial(n)
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>>> conjugate(jacobi(n, a, b, x))
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jacobi(n, conjugate(a), conjugate(b), conjugate(x))
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>>> diff(jacobi(n,a,b,x), x)
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(a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)
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See Also
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========
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gegenbauer,
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chebyshevt_root, chebyshevu, chebyshevu_root,
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legendre, assoc_legendre,
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hermite, hermite_prob,
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laguerre, assoc_laguerre,
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sympy.polys.orthopolys.jacobi_poly,
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sympy.polys.orthopolys.gegenbauer_poly
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sympy.polys.orthopolys.chebyshevt_poly
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sympy.polys.orthopolys.chebyshevu_poly
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sympy.polys.orthopolys.hermite_poly
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sympy.polys.orthopolys.legendre_poly
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sympy.polys.orthopolys.laguerre_poly
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
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.. [2] https://mathworld.wolfram.com/JacobiPolynomial.html
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.. [3] https://functions.wolfram.com/Polynomials/JacobiP/
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"""
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@classmethod
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def eval(cls, n, a, b, x):
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# Simplify to other polynomials
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# P^{a, a}_n(x)
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if a == b:
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if a == Rational(-1, 2):
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return RisingFactorial(S.Half, n) / factorial(n) * chebyshevt(n, x)
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elif a.is_zero:
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return legendre(n, x)
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elif a == S.Half:
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return RisingFactorial(3*S.Half, n) / factorial(n + 1) * chebyshevu(n, x)
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else:
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return RisingFactorial(a + 1, n) / RisingFactorial(2*a + 1, n) * gegenbauer(n, a + S.Half, x)
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elif b == -a:
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# P^{a, -a}_n(x)
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return gamma(n + a + 1) / gamma(n + 1) * (1 + x)**(a/2) / (1 - x)**(a/2) * assoc_legendre(n, -a, x)
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if not n.is_Number:
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# Symbolic result P^{a,b}_n(x)
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# P^{a,b}_n(-x) ---> (-1)**n * P^{b,a}_n(-x)
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if x.could_extract_minus_sign():
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return S.NegativeOne**n * jacobi(n, b, a, -x)
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# We can evaluate for some special values of x
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if x.is_zero:
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return (2**(-n) * gamma(a + n + 1) / (gamma(a + 1) * factorial(n)) *
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hyper([-b - n, -n], [a + 1], -1))
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if x == S.One:
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return RisingFactorial(a + 1, n) / factorial(n)
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elif x is S.Infinity:
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if n.is_positive:
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# Make sure a+b+2*n \notin Z
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if (a + b + 2*n).is_integer:
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raise ValueError("Error. a + b + 2*n should not be an integer.")
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return RisingFactorial(a + b + n + 1, n) * S.Infinity
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else:
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# n is a given fixed integer, evaluate into polynomial
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return jacobi_poly(n, a, b, x)
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def fdiff(self, argindex=4):
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from sympy.concrete.summations import Sum
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if argindex == 1:
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# Diff wrt n
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raise ArgumentIndexError(self, argindex)
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elif argindex == 2:
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# Diff wrt a
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n, a, b, x = self.args
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k = Dummy("k")
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f1 = 1 / (a + b + n + k + 1)
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f2 = ((a + b + 2*k + 1) * RisingFactorial(b + k + 1, n - k) /
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((n - k) * RisingFactorial(a + b + k + 1, n - k)))
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return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1))
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elif argindex == 3:
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# Diff wrt b
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n, a, b, x = self.args
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k = Dummy("k")
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f1 = 1 / (a + b + n + k + 1)
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f2 = (-1)**(n - k) * ((a + b + 2*k + 1) * RisingFactorial(a + k + 1, n - k) /
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((n - k) * RisingFactorial(a + b + k + 1, n - k)))
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return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1))
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elif argindex == 4:
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# Diff wrt x
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n, a, b, x = self.args
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return S.Half * (a + b + n + 1) * jacobi(n - 1, a + 1, b + 1, x)
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else:
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raise ArgumentIndexError(self, argindex)
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def _eval_rewrite_as_Sum(self, n, a, b, x, **kwargs):
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from sympy.concrete.summations import Sum
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# Make sure n \in N
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if n.is_negative or n.is_integer is False:
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raise ValueError("Error: n should be a non-negative integer.")
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k = Dummy("k")
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kern = (RisingFactorial(-n, k) * RisingFactorial(a + b + n + 1, k) * RisingFactorial(a + k + 1, n - k) /
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factorial(k) * ((1 - x)/2)**k)
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return 1 / factorial(n) * Sum(kern, (k, 0, n))
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def _eval_rewrite_as_polynomial(self, n, a, b, x, **kwargs):
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# This function is just kept for backwards compatibility
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# but should not be used
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return self._eval_rewrite_as_Sum(n, a, b, x, **kwargs)
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def _eval_conjugate(self):
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n, a, b, x = self.args
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return self.func(n, a.conjugate(), b.conjugate(), x.conjugate())
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def jacobi_normalized(n, a, b, x):
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r"""
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Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.
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Explanation
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===========
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``jacobi_normalized(n, alpha, beta, x)`` gives the $n$th
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Jacobi polynomial in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$.
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The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
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to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.
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This functions returns the polynomials normilzed:
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.. math::
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\int_{-1}^{1}
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P_m^{\left(\alpha, \beta\right)}(x)
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P_n^{\left(\alpha, \beta\right)}(x)
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(1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x
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= \delta_{m,n}
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Examples
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========
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>>> from sympy import jacobi_normalized
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>>> from sympy.abc import n,a,b,x
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>>> jacobi_normalized(n, a, b, x)
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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)))
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Parameters
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==========
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n : integer degree of polynomial
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a : alpha value
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b : beta value
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x : symbol
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See Also
|
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|
|
========
|
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|
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gegenbauer,
|
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chebyshevt_root, chebyshevu, chebyshevu_root,
|
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legendre, assoc_legendre,
|
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hermite, hermite_prob,
|
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laguerre, assoc_laguerre,
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sympy.polys.orthopolys.jacobi_poly,
|
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sympy.polys.orthopolys.gegenbauer_poly
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sympy.polys.orthopolys.chebyshevt_poly
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sympy.polys.orthopolys.chebyshevu_poly
|
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sympy.polys.orthopolys.hermite_poly
|
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sympy.polys.orthopolys.legendre_poly
|
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|
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sympy.polys.orthopolys.laguerre_poly
|
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|
|
|
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|
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References
|
||
|
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==========
|
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|
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|
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|
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.. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
|
||
|
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.. [2] https://mathworld.wolfram.com/JacobiPolynomial.html
|
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.. [3] https://functions.wolfram.com/Polynomials/JacobiP/
|
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"""
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nfactor = (S(2)**(a + b + 1) * (gamma(n + a + 1) * gamma(n + b + 1))
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/ (2*n + a + b + 1) / (factorial(n) * gamma(n + a + b + 1)))
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return jacobi(n, a, b, x) / sqrt(nfactor)
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#----------------------------------------------------------------------------
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# Gegenbauer polynomials
|
||
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#
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||
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|
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class gegenbauer(OrthogonalPolynomial):
|
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r"""
|
||
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Gegenbauer polynomial $C_n^{\left(\alpha\right)}(x)$.
|
||
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||
|
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Explanation
|
||
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|
===========
|
||
|
|
|
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``gegenbauer(n, alpha, x)`` gives the $n$th Gegenbauer polynomial
|
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in $x$, $C_n^{\left(\alpha\right)}(x)$.
|
||
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The Gegenbauer polynomials are orthogonal on $[-1, 1]$ with
|
||
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respect to the weight $\left(1-x^2\right)^{\alpha-\frac{1}{2}}$.
|
||
|
|
|
||
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|
Examples
|
||
|
|
========
|
||
|
|
|
||
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>>> from sympy import gegenbauer, conjugate, diff
|
||
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>>> from sympy.abc import n,a,x
|
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>>> gegenbauer(0, a, x)
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1
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>>> gegenbauer(1, a, x)
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2*a*x
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>>> gegenbauer(2, a, x)
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-a + x**2*(2*a**2 + 2*a)
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>>> gegenbauer(3, a, x)
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||
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x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
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>>> gegenbauer(n, a, x)
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||
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gegenbauer(n, a, x)
|
||
|
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>>> gegenbauer(n, a, -x)
|
||
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(-1)**n*gegenbauer(n, a, x)
|
||
|
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|
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>>> gegenbauer(n, a, 0)
|
||
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2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1))
|
||
|
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>>> gegenbauer(n, a, 1)
|
||
|
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gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))
|
||
|
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|
||
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>>> conjugate(gegenbauer(n, a, x))
|
||
|
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gegenbauer(n, conjugate(a), conjugate(x))
|
||
|
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|
||
|
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>>> 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, hermite_prob,
|
||
|
|
laguerre, assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.polys.orthopolys.laguerre_poly
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials
|
||
|
|
.. [2] https://mathworld.wolfram.com/GegenbauerPolynomial.html
|
||
|
|
.. [3] https://functions.wolfram.com/Polynomials/GegenbauerC3/
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def eval(cls, n, a, x):
|
||
|
|
# For negative n the polynomials vanish
|
||
|
|
# See https://functions.wolfram.com/Polynomials/GegenbauerC3/03/01/03/0012/
|
||
|
|
if n.is_negative:
|
||
|
|
return S.Zero
|
||
|
|
|
||
|
|
# Some special values for fixed a
|
||
|
|
if a == S.Half:
|
||
|
|
return legendre(n, x)
|
||
|
|
elif a == S.One:
|
||
|
|
return chebyshevu(n, x)
|
||
|
|
elif a == S.NegativeOne:
|
||
|
|
return S.Zero
|
||
|
|
|
||
|
|
if not n.is_Number:
|
||
|
|
# Handle this before the general sign extraction rule
|
||
|
|
if x == S.NegativeOne:
|
||
|
|
if (re(a) > S.Half) == True:
|
||
|
|
return S.ComplexInfinity
|
||
|
|
else:
|
||
|
|
return (cos(S.Pi*(a+n)) * sec(S.Pi*a) * gamma(2*a+n) /
|
||
|
|
(gamma(2*a) * gamma(n+1)))
|
||
|
|
|
||
|
|
# Symbolic result C^a_n(x)
|
||
|
|
# C^a_n(-x) ---> (-1)**n * C^a_n(x)
|
||
|
|
if x.could_extract_minus_sign():
|
||
|
|
return S.NegativeOne**n * gegenbauer(n, a, -x)
|
||
|
|
# We can evaluate for some special values of x
|
||
|
|
if x.is_zero:
|
||
|
|
return (2**n * sqrt(S.Pi) * gamma(a + S.Half*n) /
|
||
|
|
(gamma((1 - n)/2) * gamma(n + 1) * gamma(a)) )
|
||
|
|
if x == S.One:
|
||
|
|
return gamma(2*a + n) / (gamma(2*a) * gamma(n + 1))
|
||
|
|
elif x is S.Infinity:
|
||
|
|
if n.is_positive:
|
||
|
|
return RisingFactorial(a, n) * S.Infinity
|
||
|
|
else:
|
||
|
|
# n is a given fixed integer, evaluate into polynomial
|
||
|
|
return gegenbauer_poly(n, a, x)
|
||
|
|
|
||
|
|
def fdiff(self, argindex=3):
|
||
|
|
from sympy.concrete.summations 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_rewrite_as_Sum(self, n, a, x, **kwargs):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
k = Dummy("k")
|
||
|
|
kern = ((-1)**k * RisingFactorial(a, n - k) * (2*x)**(n - 2*k) /
|
||
|
|
(factorial(k) * factorial(n - 2*k)))
|
||
|
|
return Sum(kern, (k, 0, floor(n/2)))
|
||
|
|
|
||
|
|
def _eval_rewrite_as_polynomial(self, n, a, x, **kwargs):
|
||
|
|
# This function is just kept for backwards compatibility
|
||
|
|
# but should not be used
|
||
|
|
return self._eval_rewrite_as_Sum(n, a, x, **kwargs)
|
||
|
|
|
||
|
|
def _eval_conjugate(self):
|
||
|
|
n, a, x = self.args
|
||
|
|
return self.func(n, a.conjugate(), x.conjugate())
|
||
|
|
|
||
|
|
#----------------------------------------------------------------------------
|
||
|
|
# Chebyshev polynomials of first and second kind
|
||
|
|
#
|
||
|
|
|
||
|
|
|
||
|
|
class chebyshevt(OrthogonalPolynomial):
|
||
|
|
r"""
|
||
|
|
Chebyshev polynomial of the first kind, $T_n(x)$.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
``chebyshevt(n, x)`` gives the $n$th 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
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import chebyshevt, diff
|
||
|
|
>>> from sympy.abc import n,x
|
||
|
|
>>> 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, hermite_prob,
|
||
|
|
laguerre, assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.polys.orthopolys.laguerre_poly
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
|
||
|
|
.. [2] https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
|
||
|
|
.. [3] https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
|
||
|
|
.. [4] https://functions.wolfram.com/Polynomials/ChebyshevT/
|
||
|
|
.. [5] https://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 S.NegativeOne**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.is_zero:
|
||
|
|
return cos(S.Half * S.Pi * n)
|
||
|
|
if x == S.One:
|
||
|
|
return S.One
|
||
|
|
elif x is S.Infinity:
|
||
|
|
return S.Infinity
|
||
|
|
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 == 1:
|
||
|
|
# Diff wrt n
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
elif argindex == 2:
|
||
|
|
# Diff wrt x
|
||
|
|
n, x = self.args
|
||
|
|
return n * chebyshevu(n - 1, x)
|
||
|
|
else:
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_Sum(self, n, x, **kwargs):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
k = Dummy("k")
|
||
|
|
kern = binomial(n, 2*k) * (x**2 - 1)**k * x**(n - 2*k)
|
||
|
|
return Sum(kern, (k, 0, floor(n/2)))
|
||
|
|
|
||
|
|
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
|
||
|
|
# This function is just kept for backwards compatibility
|
||
|
|
# but should not be used
|
||
|
|
return self._eval_rewrite_as_Sum(n, x, **kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
class chebyshevu(OrthogonalPolynomial):
|
||
|
|
r"""
|
||
|
|
Chebyshev polynomial of the second kind, $U_n(x)$.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
``chebyshevu(n, x)`` gives the $n$th 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
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import chebyshevu, diff
|
||
|
|
>>> from sympy.abc import n,x
|
||
|
|
>>> 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, hermite_prob,
|
||
|
|
laguerre, assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.polys.orthopolys.laguerre_poly
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
|
||
|
|
.. [2] https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
|
||
|
|
.. [3] https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
|
||
|
|
.. [4] https://functions.wolfram.com/Polynomials/ChebyshevT/
|
||
|
|
.. [5] https://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 S.NegativeOne**n * chebyshevu(n, -x)
|
||
|
|
# U_{-n}(x) ---> -U_{n-2}(x)
|
||
|
|
if n.could_extract_minus_sign():
|
||
|
|
if n == S.NegativeOne:
|
||
|
|
# n can not be -1 here
|
||
|
|
return S.Zero
|
||
|
|
elif not (-n - 2).could_extract_minus_sign():
|
||
|
|
return -chebyshevu(-n - 2, x)
|
||
|
|
# We can evaluate for some special values of x
|
||
|
|
if x.is_zero:
|
||
|
|
return cos(S.Half * S.Pi * n)
|
||
|
|
if x == S.One:
|
||
|
|
return S.One + n
|
||
|
|
elif x is S.Infinity:
|
||
|
|
return S.Infinity
|
||
|
|
else:
|
||
|
|
# n is a given fixed integer, evaluate into polynomial
|
||
|
|
if n.is_negative:
|
||
|
|
# U_{-n}(x) ---> -U_{n-2}(x)
|
||
|
|
if n == S.NegativeOne:
|
||
|
|
return S.Zero
|
||
|
|
else:
|
||
|
|
return -cls._eval_at_order(-n - 2, x)
|
||
|
|
else:
|
||
|
|
return cls._eval_at_order(n, x)
|
||
|
|
|
||
|
|
def fdiff(self, argindex=2):
|
||
|
|
if argindex == 1:
|
||
|
|
# Diff wrt n
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
elif argindex == 2:
|
||
|
|
# Diff wrt x
|
||
|
|
n, x = self.args
|
||
|
|
return ((n + 1) * chebyshevt(n + 1, x) - x * chebyshevu(n, x)) / (x**2 - 1)
|
||
|
|
else:
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_Sum(self, n, x, **kwargs):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
k = Dummy("k")
|
||
|
|
kern = S.NegativeOne**k * factorial(
|
||
|
|
n - k) * (2*x)**(n - 2*k) / (factorial(k) * factorial(n - 2*k))
|
||
|
|
return Sum(kern, (k, 0, floor(n/2)))
|
||
|
|
|
||
|
|
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
|
||
|
|
# This function is just kept for backwards compatibility
|
||
|
|
# but should not be used
|
||
|
|
return self._eval_rewrite_as_Sum(n, x, **kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
class chebyshevt_root(Function):
|
||
|
|
r"""
|
||
|
|
``chebyshev_root(n, k)`` returns the $k$th root (indexed from zero) of
|
||
|
|
the $n$th Chebyshev polynomial of the first kind; that is, if
|
||
|
|
$0 \le k < n$, ``chebyshevt(n, chebyshevt_root(n, k)) == 0``.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import chebyshevt, chebyshevt_root
|
||
|
|
>>> 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, hermite_prob,
|
||
|
|
laguerre, assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.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(S.Pi*(2*k + 1)/(2*n))
|
||
|
|
|
||
|
|
|
||
|
|
class chebyshevu_root(Function):
|
||
|
|
r"""
|
||
|
|
``chebyshevu_root(n, k)`` returns the $k$th root (indexed from zero) of the
|
||
|
|
$n$th Chebyshev polynomial of the second kind; that is, if $0 \le k < n$,
|
||
|
|
``chebyshevu(n, chebyshevu_root(n, k)) == 0``.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import chebyshevu, chebyshevu_root
|
||
|
|
>>> chebyshevu_root(3, 2)
|
||
|
|
-sqrt(2)/2
|
||
|
|
>>> chebyshevu(3, chebyshevu_root(3, 2))
|
||
|
|
0
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
chebyshevt, chebyshevt_root, chebyshevu,
|
||
|
|
legendre, assoc_legendre,
|
||
|
|
hermite, hermite_prob,
|
||
|
|
laguerre, assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.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(S.Pi*(k + 1)/(n + 1))
|
||
|
|
|
||
|
|
#----------------------------------------------------------------------------
|
||
|
|
# Legendre polynomials and Associated Legendre polynomials
|
||
|
|
#
|
||
|
|
|
||
|
|
|
||
|
|
class legendre(OrthogonalPolynomial):
|
||
|
|
r"""
|
||
|
|
``legendre(n, x)`` gives the $n$th Legendre polynomial of $x$, $P_n(x)$
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
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
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import legendre, diff
|
||
|
|
>>> from sympy.abc import x, n
|
||
|
|
>>> 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, hermite_prob,
|
||
|
|
laguerre, assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.polys.orthopolys.laguerre_poly
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Legendre_polynomial
|
||
|
|
.. [2] https://mathworld.wolfram.com/LegendrePolynomial.html
|
||
|
|
.. [3] https://functions.wolfram.com/Polynomials/LegendreP/
|
||
|
|
.. [4] https://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 S.NegativeOne**n * legendre(n, -x)
|
||
|
|
# L_{-n}(x) ---> L_{n-1}(x)
|
||
|
|
if n.could_extract_minus_sign() and not(-n - 1).could_extract_minus_sign():
|
||
|
|
return legendre(-n - S.One, x)
|
||
|
|
# We can evaluate for some special values of x
|
||
|
|
if x.is_zero:
|
||
|
|
return sqrt(S.Pi)/(gamma(S.Half - n/2)*gamma(S.One + n/2))
|
||
|
|
elif x == S.One:
|
||
|
|
return S.One
|
||
|
|
elif x is S.Infinity:
|
||
|
|
return S.Infinity
|
||
|
|
else:
|
||
|
|
# n is a given fixed integer, evaluate into polynomial;
|
||
|
|
# L_{-n}(x) ---> L_{n-1}(x)
|
||
|
|
if n.is_negative:
|
||
|
|
n = -n - S.One
|
||
|
|
return cls._eval_at_order(n, x)
|
||
|
|
|
||
|
|
def fdiff(self, argindex=2):
|
||
|
|
if argindex == 1:
|
||
|
|
# Diff wrt n
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
elif argindex == 2:
|
||
|
|
# Diff wrt x
|
||
|
|
# Find better formula, this is unsuitable for x = +/-1
|
||
|
|
# https://www.autodiff.org/ad16/Oral/Buecker_Legendre.pdf says
|
||
|
|
# at x = 1:
|
||
|
|
# n*(n + 1)/2 , m = 0
|
||
|
|
# oo , m = 1
|
||
|
|
# -(n-1)*n*(n+1)*(n+2)/4 , m = 2
|
||
|
|
# 0 , m = 3, 4, ..., n
|
||
|
|
#
|
||
|
|
# at x = -1
|
||
|
|
# (-1)**(n+1)*n*(n + 1)/2 , m = 0
|
||
|
|
# (-1)**n*oo , m = 1
|
||
|
|
# (-1)**n*(n-1)*n*(n+1)*(n+2)/4 , m = 2
|
||
|
|
# 0 , m = 3, 4, ..., n
|
||
|
|
n, x = self.args
|
||
|
|
return n/(x**2 - 1)*(x*legendre(n, x) - legendre(n - 1, x))
|
||
|
|
else:
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_Sum(self, n, x, **kwargs):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
k = Dummy("k")
|
||
|
|
kern = S.NegativeOne**k*binomial(n, k)**2*((1 + x)/2)**(n - k)*((1 - x)/2)**k
|
||
|
|
return Sum(kern, (k, 0, n))
|
||
|
|
|
||
|
|
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
|
||
|
|
# This function is just kept for backwards compatibility
|
||
|
|
# but should not be used
|
||
|
|
return self._eval_rewrite_as_Sum(n, x, **kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
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}
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
Associated Legendre polynomials are orthogonal on $[-1, 1]$ with:
|
||
|
|
|
||
|
|
- weight $= 1$ for the same $m$ and different $n$.
|
||
|
|
- weight $= \frac{1}{1-x^2}$ for the same $n$ and different $m$.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import assoc_legendre
|
||
|
|
>>> from sympy.abc import x, m, n
|
||
|
|
>>> assoc_legendre(0,0, x)
|
||
|
|
1
|
||
|
|
>>> assoc_legendre(1,0, x)
|
||
|
|
x
|
||
|
|
>>> assoc_legendre(1,1, x)
|
||
|
|
-sqrt(1 - x**2)
|
||
|
|
>>> assoc_legendre(n,m,x)
|
||
|
|
assoc_legendre(n, m, x)
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
jacobi, gegenbauer,
|
||
|
|
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
|
||
|
|
legendre,
|
||
|
|
hermite, hermite_prob,
|
||
|
|
laguerre, assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.polys.orthopolys.laguerre_poly
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
|
||
|
|
.. [2] https://mathworld.wolfram.com/LegendrePolynomial.html
|
||
|
|
.. [3] https://functions.wolfram.com/Polynomials/LegendreP/
|
||
|
|
.. [4] https://functions.wolfram.com/Polynomials/LegendreP2/
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def _eval_at_order(cls, n, m):
|
||
|
|
P = legendre_poly(n, _x, polys=True).diff((_x, m))
|
||
|
|
return S.NegativeOne**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 S.NegativeOne**(-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(S.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 == 1:
|
||
|
|
# Diff wrt n
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
elif argindex == 2:
|
||
|
|
# Diff wrt m
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
elif 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:
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_Sum(self, n, m, x, **kwargs):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
k = Dummy("k")
|
||
|
|
kern = factorial(2*n - 2*k)/(2**n*factorial(n - k)*factorial(
|
||
|
|
k)*factorial(n - 2*k - m))*S.NegativeOne**k*x**(n - m - 2*k)
|
||
|
|
return (1 - x**2)**(m/2) * Sum(kern, (k, 0, floor((n - m)*S.Half)))
|
||
|
|
|
||
|
|
def _eval_rewrite_as_polynomial(self, n, m, x, **kwargs):
|
||
|
|
# This function is just kept for backwards compatibility
|
||
|
|
# but should not be used
|
||
|
|
return self._eval_rewrite_as_Sum(n, m, x, **kwargs)
|
||
|
|
|
||
|
|
def _eval_conjugate(self):
|
||
|
|
n, m, x = self.args
|
||
|
|
return self.func(n, m.conjugate(), x.conjugate())
|
||
|
|
|
||
|
|
#----------------------------------------------------------------------------
|
||
|
|
# Hermite polynomials
|
||
|
|
#
|
||
|
|
|
||
|
|
|
||
|
|
class hermite(OrthogonalPolynomial):
|
||
|
|
r"""
|
||
|
|
``hermite(n, x)`` gives the $n$th Hermite polynomial in $x$, $H_n(x)$.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
The Hermite polynomials are orthogonal on $(-\infty, \infty)$
|
||
|
|
with respect to the weight $\exp\left(-x^2\right)$.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import hermite, diff
|
||
|
|
>>> from sympy.abc import x, n
|
||
|
|
>>> 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,
|
||
|
|
hermite_prob,
|
||
|
|
laguerre, assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.polys.orthopolys.laguerre_poly
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Hermite_polynomial
|
||
|
|
.. [2] https://mathworld.wolfram.com/HermitePolynomial.html
|
||
|
|
.. [3] https://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 S.NegativeOne**n * hermite(n, -x)
|
||
|
|
# We can evaluate for some special values of x
|
||
|
|
if x.is_zero:
|
||
|
|
return 2**n * sqrt(S.Pi) / gamma((S.One - n)/2)
|
||
|
|
elif x is S.Infinity:
|
||
|
|
return S.Infinity
|
||
|
|
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 == 1:
|
||
|
|
# Diff wrt n
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
elif argindex == 2:
|
||
|
|
# Diff wrt x
|
||
|
|
n, x = self.args
|
||
|
|
return 2*n*hermite(n - 1, x)
|
||
|
|
else:
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_Sum(self, n, x, **kwargs):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
k = Dummy("k")
|
||
|
|
kern = S.NegativeOne**k / (factorial(k)*factorial(n - 2*k)) * (2*x)**(n - 2*k)
|
||
|
|
return factorial(n)*Sum(kern, (k, 0, floor(n/2)))
|
||
|
|
|
||
|
|
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
|
||
|
|
# This function is just kept for backwards compatibility
|
||
|
|
# but should not be used
|
||
|
|
return self._eval_rewrite_as_Sum(n, x, **kwargs)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_hermite_prob(self, n, x, **kwargs):
|
||
|
|
return sqrt(2)**n * hermite_prob(n, x*sqrt(2))
|
||
|
|
|
||
|
|
|
||
|
|
class hermite_prob(OrthogonalPolynomial):
|
||
|
|
r"""
|
||
|
|
``hermite_prob(n, x)`` gives the $n$th probabilist's Hermite polynomial
|
||
|
|
in $x$, $He_n(x)$.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
The probabilist's Hermite polynomials are orthogonal on $(-\infty, \infty)$
|
||
|
|
with respect to the weight $\exp\left(-\frac{x^2}{2}\right)$. They are monic
|
||
|
|
polynomials, related to the plain Hermite polynomials (:py:class:`~.hermite`) by
|
||
|
|
|
||
|
|
.. math :: He_n(x) = 2^{-n/2} H_n(x/\sqrt{2})
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import hermite_prob, diff, I
|
||
|
|
>>> from sympy.abc import x, n
|
||
|
|
>>> hermite_prob(1, x)
|
||
|
|
x
|
||
|
|
>>> hermite_prob(5, x)
|
||
|
|
x**5 - 10*x**3 + 15*x
|
||
|
|
>>> diff(hermite_prob(n,x), x)
|
||
|
|
n*hermite_prob(n - 1, x)
|
||
|
|
>>> hermite_prob(n, -x)
|
||
|
|
(-1)**n*hermite_prob(n, x)
|
||
|
|
|
||
|
|
The sum of absolute values of coefficients of $He_n(x)$ is the number of
|
||
|
|
matchings in the complete graph $K_n$ or telephone number, A000085 in the OEIS:
|
||
|
|
|
||
|
|
>>> [hermite_prob(n,I) / I**n for n in range(11)]
|
||
|
|
[1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
jacobi, gegenbauer,
|
||
|
|
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
|
||
|
|
legendre, assoc_legendre,
|
||
|
|
hermite,
|
||
|
|
laguerre, assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.polys.orthopolys.laguerre_poly
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Hermite_polynomial
|
||
|
|
.. [2] https://mathworld.wolfram.com/HermitePolynomial.html
|
||
|
|
"""
|
||
|
|
|
||
|
|
_ortho_poly = staticmethod(hermite_prob_poly)
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def eval(cls, n, x):
|
||
|
|
if not n.is_Number:
|
||
|
|
if x.could_extract_minus_sign():
|
||
|
|
return S.NegativeOne**n * hermite_prob(n, -x)
|
||
|
|
if x.is_zero:
|
||
|
|
return sqrt(S.Pi) / gamma((S.One-n) / 2)
|
||
|
|
elif x is S.Infinity:
|
||
|
|
return S.Infinity
|
||
|
|
else:
|
||
|
|
if n.is_negative:
|
||
|
|
ValueError("n must be a nonnegative integer, not %r" % n)
|
||
|
|
else:
|
||
|
|
return cls._eval_at_order(n, x)
|
||
|
|
|
||
|
|
def fdiff(self, argindex=2):
|
||
|
|
if argindex == 2:
|
||
|
|
n, x = self.args
|
||
|
|
return n*hermite_prob(n-1, x)
|
||
|
|
else:
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_Sum(self, n, x, **kwargs):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
k = Dummy("k")
|
||
|
|
kern = (-S.Half)**k * x**(n-2*k) / (factorial(k) * factorial(n-2*k))
|
||
|
|
return factorial(n)*Sum(kern, (k, 0, floor(n/2)))
|
||
|
|
|
||
|
|
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
|
||
|
|
# This function is just kept for backwards compatibility
|
||
|
|
# but should not be used
|
||
|
|
return self._eval_rewrite_as_Sum(n, x, **kwargs)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_hermite(self, n, x, **kwargs):
|
||
|
|
return sqrt(2)**(-n) * hermite(n, x/sqrt(2))
|
||
|
|
|
||
|
|
|
||
|
|
#----------------------------------------------------------------------------
|
||
|
|
# Laguerre polynomials
|
||
|
|
#
|
||
|
|
|
||
|
|
|
||
|
|
class laguerre(OrthogonalPolynomial):
|
||
|
|
r"""
|
||
|
|
Returns the $n$th Laguerre polynomial in $x$, $L_n(x)$.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import laguerre, diff
|
||
|
|
>>> from sympy.abc import x, n
|
||
|
|
>>> laguerre(0, x)
|
||
|
|
1
|
||
|
|
>>> laguerre(1, x)
|
||
|
|
1 - x
|
||
|
|
>>> 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)
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
n : int
|
||
|
|
Degree of Laguerre polynomial. Must be `n \ge 0`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
jacobi, gegenbauer,
|
||
|
|
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
|
||
|
|
legendre, assoc_legendre,
|
||
|
|
hermite, hermite_prob,
|
||
|
|
assoc_laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.polys.orthopolys.laguerre_poly
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial
|
||
|
|
.. [2] https://mathworld.wolfram.com/LaguerrePolynomial.html
|
||
|
|
.. [3] https://functions.wolfram.com/Polynomials/LaguerreL/
|
||
|
|
.. [4] https://functions.wolfram.com/Polynomials/LaguerreL3/
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
_ortho_poly = staticmethod(laguerre_poly)
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def eval(cls, n, x):
|
||
|
|
if n.is_integer is False:
|
||
|
|
raise ValueError("Error: n should be an integer.")
|
||
|
|
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() and not(-n - 1).could_extract_minus_sign():
|
||
|
|
return exp(x)*laguerre(-n - 1, -x)
|
||
|
|
# We can evaluate for some special values of x
|
||
|
|
if x.is_zero:
|
||
|
|
return S.One
|
||
|
|
elif x is S.NegativeInfinity:
|
||
|
|
return S.Infinity
|
||
|
|
elif x is S.Infinity:
|
||
|
|
return S.NegativeOne**n * S.Infinity
|
||
|
|
else:
|
||
|
|
if n.is_negative:
|
||
|
|
return exp(x)*laguerre(-n - 1, -x)
|
||
|
|
else:
|
||
|
|
return cls._eval_at_order(n, x)
|
||
|
|
|
||
|
|
def fdiff(self, argindex=2):
|
||
|
|
if argindex == 1:
|
||
|
|
# Diff wrt n
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
elif argindex == 2:
|
||
|
|
# Diff wrt x
|
||
|
|
n, x = self.args
|
||
|
|
return -assoc_laguerre(n - 1, 1, x)
|
||
|
|
else:
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_Sum(self, n, x, **kwargs):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
# Make sure n \in N_0
|
||
|
|
if n.is_negative:
|
||
|
|
return exp(x) * self._eval_rewrite_as_Sum(-n - 1, -x, **kwargs)
|
||
|
|
if n.is_integer is False:
|
||
|
|
raise ValueError("Error: n should be an integer.")
|
||
|
|
k = Dummy("k")
|
||
|
|
kern = RisingFactorial(-n, k) / factorial(k)**2 * x**k
|
||
|
|
return Sum(kern, (k, 0, n))
|
||
|
|
|
||
|
|
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
|
||
|
|
# This function is just kept for backwards compatibility
|
||
|
|
# but should not be used
|
||
|
|
return self._eval_rewrite_as_Sum(n, x, **kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
class assoc_laguerre(OrthogonalPolynomial):
|
||
|
|
r"""
|
||
|
|
Returns the $n$th generalized Laguerre polynomial in $x$, $L_n(x)$.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import assoc_laguerre, diff
|
||
|
|
>>> from sympy.abc import x, n, a
|
||
|
|
>>> 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))
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
n : int
|
||
|
|
Degree of Laguerre polynomial. Must be `n \ge 0`.
|
||
|
|
|
||
|
|
alpha : Expr
|
||
|
|
Arbitrary expression. For ``alpha=0`` regular Laguerre
|
||
|
|
polynomials will be generated.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
========
|
||
|
|
|
||
|
|
jacobi, gegenbauer,
|
||
|
|
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
|
||
|
|
legendre, assoc_legendre,
|
||
|
|
hermite, hermite_prob,
|
||
|
|
laguerre,
|
||
|
|
sympy.polys.orthopolys.jacobi_poly
|
||
|
|
sympy.polys.orthopolys.gegenbauer_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevt_poly
|
||
|
|
sympy.polys.orthopolys.chebyshevu_poly
|
||
|
|
sympy.polys.orthopolys.hermite_poly
|
||
|
|
sympy.polys.orthopolys.hermite_prob_poly
|
||
|
|
sympy.polys.orthopolys.legendre_poly
|
||
|
|
sympy.polys.orthopolys.laguerre_poly
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Generalized_Laguerre_polynomials
|
||
|
|
.. [2] https://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
|
||
|
|
.. [3] https://functions.wolfram.com/Polynomials/LaguerreL/
|
||
|
|
.. [4] https://functions.wolfram.com/Polynomials/LaguerreL3/
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def eval(cls, n, alpha, x):
|
||
|
|
# L_{n}^{0}(x) ---> L_{n}(x)
|
||
|
|
if alpha.is_zero:
|
||
|
|
return laguerre(n, x)
|
||
|
|
|
||
|
|
if not n.is_Number:
|
||
|
|
# We can evaluate for some special values of x
|
||
|
|
if x.is_zero:
|
||
|
|
return binomial(n + alpha, alpha)
|
||
|
|
elif x is S.Infinity and n > 0:
|
||
|
|
return S.NegativeOne**n * S.Infinity
|
||
|
|
elif x is S.NegativeInfinity and n > 0:
|
||
|
|
return S.Infinity
|
||
|
|
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 laguerre_poly(n, x, alpha)
|
||
|
|
|
||
|
|
def fdiff(self, argindex=3):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
if argindex == 1:
|
||
|
|
# Diff wrt n
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
elif 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:
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_Sum(self, n, alpha, x, **kwargs):
|
||
|
|
from sympy.concrete.summations import Sum
|
||
|
|
# Make sure n \in N_0
|
||
|
|
if n.is_negative or n.is_integer is False:
|
||
|
|
raise ValueError("Error: n should be a non-negative integer.")
|
||
|
|
k = Dummy("k")
|
||
|
|
kern = RisingFactorial(
|
||
|
|
-n, k) / (gamma(k + alpha + 1) * factorial(k)) * x**k
|
||
|
|
return gamma(n + alpha + 1) / factorial(n) * Sum(kern, (k, 0, n))
|
||
|
|
|
||
|
|
def _eval_rewrite_as_polynomial(self, n, alpha, x, **kwargs):
|
||
|
|
# This function is just kept for backwards compatibility
|
||
|
|
# but should not be used
|
||
|
|
return self._eval_rewrite_as_Sum(n, alpha, x, **kwargs)
|
||
|
|
|
||
|
|
def _eval_conjugate(self):
|
||
|
|
n, alpha, x = self.args
|
||
|
|
return self.func(n, alpha.conjugate(), x.conjugate())
|