419 lines
11 KiB
Python
419 lines
11 KiB
Python
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"""This module implements tools for integrating rational functions. """
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from sympy.core.function import Lambda
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from sympy.core.numbers import I
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from sympy.core.singleton import S
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from sympy.core.symbol import (Dummy, Symbol, symbols)
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from sympy.functions.elementary.exponential import log
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from sympy.functions.elementary.trigonometric import atan
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from sympy.polys.polyroots import roots
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from sympy.polys.polytools import cancel
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from sympy.polys.rootoftools import RootSum
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from sympy.polys import Poly, resultant, ZZ
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def ratint(f, x, **flags):
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"""
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Performs indefinite integration of rational functions.
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Explanation
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===========
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Given a field :math:`K` and a rational function :math:`f = p/q`,
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where :math:`p` and :math:`q` are polynomials in :math:`K[x]`,
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returns a function :math:`g` such that :math:`f = g'`.
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Examples
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========
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>>> from sympy.integrals.rationaltools import ratint
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>>> from sympy.abc import x
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>>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
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(12*x + 6)/(x**2 - 1) + 4*log(x - 2) - 4*log(x + 1)
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References
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==========
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.. [1] M. Bronstein, Symbolic Integration I: Transcendental
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Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70
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See Also
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========
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sympy.integrals.integrals.Integral.doit
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sympy.integrals.rationaltools.ratint_logpart
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sympy.integrals.rationaltools.ratint_ratpart
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"""
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if isinstance(f, tuple):
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p, q = f
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else:
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p, q = f.as_numer_denom()
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p, q = Poly(p, x, composite=False, field=True), Poly(q, x, composite=False, field=True)
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coeff, p, q = p.cancel(q)
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poly, p = p.div(q)
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result = poly.integrate(x).as_expr()
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if p.is_zero:
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return coeff*result
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g, h = ratint_ratpart(p, q, x)
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P, Q = h.as_numer_denom()
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P = Poly(P, x)
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Q = Poly(Q, x)
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q, r = P.div(Q)
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result += g + q.integrate(x).as_expr()
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if not r.is_zero:
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symbol = flags.get('symbol', 't')
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if not isinstance(symbol, Symbol):
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t = Dummy(symbol)
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else:
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t = symbol.as_dummy()
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L = ratint_logpart(r, Q, x, t)
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real = flags.get('real')
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if real is None:
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if isinstance(f, tuple):
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p, q = f
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atoms = p.atoms() | q.atoms()
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else:
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atoms = f.atoms()
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for elt in atoms - {x}:
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if not elt.is_extended_real:
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real = False
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break
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else:
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real = True
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eps = S.Zero
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if not real:
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for h, q in L:
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_, h = h.primitive()
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eps += RootSum(
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q, Lambda(t, t*log(h.as_expr())), quadratic=True)
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else:
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for h, q in L:
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_, h = h.primitive()
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R = log_to_real(h, q, x, t)
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if R is not None:
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eps += R
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else:
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eps += RootSum(
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q, Lambda(t, t*log(h.as_expr())), quadratic=True)
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result += eps
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return coeff*result
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def ratint_ratpart(f, g, x):
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"""
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Horowitz-Ostrogradsky algorithm.
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Explanation
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===========
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Given a field K and polynomials f and g in K[x], such that f and g
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are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
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such that f/g = A' + B and B has square-free denominator.
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Examples
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========
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>>> from sympy.integrals.rationaltools import ratint_ratpart
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>>> from sympy.abc import x, y
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>>> from sympy import Poly
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>>> ratint_ratpart(Poly(1, x, domain='ZZ'),
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... Poly(x + 1, x, domain='ZZ'), x)
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(0, 1/(x + 1))
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>>> ratint_ratpart(Poly(1, x, domain='EX'),
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... Poly(x**2 + y**2, x, domain='EX'), x)
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(0, 1/(x**2 + y**2))
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>>> ratint_ratpart(Poly(36, x, domain='ZZ'),
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... Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x, domain='ZZ'), x)
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((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2))
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See Also
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========
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ratint, ratint_logpart
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"""
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from sympy.solvers.solvers import solve
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f = Poly(f, x)
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g = Poly(g, x)
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u, v, _ = g.cofactors(g.diff())
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n = u.degree()
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m = v.degree()
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A_coeffs = [ Dummy('a' + str(n - i)) for i in range(0, n) ]
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B_coeffs = [ Dummy('b' + str(m - i)) for i in range(0, m) ]
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C_coeffs = A_coeffs + B_coeffs
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A = Poly(A_coeffs, x, domain=ZZ[C_coeffs])
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B = Poly(B_coeffs, x, domain=ZZ[C_coeffs])
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H = f - A.diff()*v + A*(u.diff()*v).quo(u) - B*u
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result = solve(H.coeffs(), C_coeffs)
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A = A.as_expr().subs(result)
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B = B.as_expr().subs(result)
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rat_part = cancel(A/u.as_expr(), x)
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log_part = cancel(B/v.as_expr(), x)
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return rat_part, log_part
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def ratint_logpart(f, g, x, t=None):
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r"""
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Lazard-Rioboo-Trager algorithm.
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Explanation
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===========
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Given a field K and polynomials f and g in K[x], such that f and g
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are coprime, deg(f) < deg(g) and g is square-free, returns a list
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of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
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in K[t, x] and q_i in K[t], and::
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___ ___
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d f d \ ` \ `
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-- - = -- ) ) a log(s_i(a, x))
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dx g dx /__, /__,
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i=1..n a | q_i(a) = 0
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Examples
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========
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>>> from sympy.integrals.rationaltools import ratint_logpart
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>>> from sympy.abc import x
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>>> from sympy import Poly
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>>> ratint_logpart(Poly(1, x, domain='ZZ'),
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... Poly(x**2 + x + 1, x, domain='ZZ'), x)
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[(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'),
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...Poly(3*_t**2 + 1, _t, domain='ZZ'))]
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>>> ratint_logpart(Poly(12, x, domain='ZZ'),
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... Poly(x**2 - x - 2, x, domain='ZZ'), x)
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[(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'),
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...Poly(-_t**2 + 16, _t, domain='ZZ'))]
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See Also
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========
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ratint, ratint_ratpart
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"""
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f, g = Poly(f, x), Poly(g, x)
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t = t or Dummy('t')
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a, b = g, f - g.diff()*Poly(t, x)
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res, R = resultant(a, b, includePRS=True)
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res = Poly(res, t, composite=False)
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assert res, "BUG: resultant(%s, %s) cannot be zero" % (a, b)
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R_map, H = {}, []
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for r in R:
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R_map[r.degree()] = r
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def _include_sign(c, sqf):
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if c.is_extended_real and (c < 0) == True:
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h, k = sqf[0]
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c_poly = c.as_poly(h.gens)
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sqf[0] = h*c_poly, k
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C, res_sqf = res.sqf_list()
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_include_sign(C, res_sqf)
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for q, i in res_sqf:
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_, q = q.primitive()
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if g.degree() == i:
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H.append((g, q))
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else:
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h = R_map[i]
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h_lc = Poly(h.LC(), t, field=True)
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c, h_lc_sqf = h_lc.sqf_list(all=True)
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_include_sign(c, h_lc_sqf)
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for a, j in h_lc_sqf:
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h = h.quo(Poly(a.gcd(q)**j, x))
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inv, coeffs = h_lc.invert(q), [S.One]
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for coeff in h.coeffs()[1:]:
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coeff = coeff.as_poly(inv.gens)
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T = (inv*coeff).rem(q)
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coeffs.append(T.as_expr())
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h = Poly(dict(list(zip(h.monoms(), coeffs))), x)
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H.append((h, q))
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return H
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def log_to_atan(f, g):
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"""
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Convert complex logarithms to real arctangents.
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Explanation
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===========
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Given a real field K and polynomials f and g in K[x], with g != 0,
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returns a sum h of arctangents of polynomials in K[x], such that:
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dh d f + I g
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-- = -- I log( ------- )
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dx dx f - I g
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Examples
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========
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>>> from sympy.integrals.rationaltools import log_to_atan
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>>> from sympy.abc import x
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>>> from sympy import Poly, sqrt, S
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>>> log_to_atan(Poly(x, x, domain='ZZ'), Poly(1, x, domain='ZZ'))
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2*atan(x)
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>>> log_to_atan(Poly(x + S(1)/2, x, domain='QQ'),
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... Poly(sqrt(3)/2, x, domain='EX'))
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2*atan(2*sqrt(3)*x/3 + sqrt(3)/3)
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See Also
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========
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log_to_real
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"""
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if f.degree() < g.degree():
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f, g = -g, f
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f = f.to_field()
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g = g.to_field()
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p, q = f.div(g)
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if q.is_zero:
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return 2*atan(p.as_expr())
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else:
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s, t, h = g.gcdex(-f)
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u = (f*s + g*t).quo(h)
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A = 2*atan(u.as_expr())
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return A + log_to_atan(s, t)
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def log_to_real(h, q, x, t):
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r"""
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Convert complex logarithms to real functions.
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Explanation
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===========
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Given real field K and polynomials h in K[t,x] and q in K[t],
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returns real function f such that:
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___
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df d \ `
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-- = -- ) a log(h(a, x))
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dx dx /__,
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a | q(a) = 0
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Examples
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========
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>>> from sympy.integrals.rationaltools import log_to_real
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>>> from sympy.abc import x, y
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>>> from sympy import Poly, S
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>>> log_to_real(Poly(x + 3*y/2 + S(1)/2, x, domain='QQ[y]'),
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... Poly(3*y**2 + 1, y, domain='ZZ'), x, y)
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2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/3
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>>> log_to_real(Poly(x**2 - 1, x, domain='ZZ'),
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... Poly(-2*y + 1, y, domain='ZZ'), x, y)
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log(x**2 - 1)/2
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See Also
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========
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log_to_atan
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"""
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from sympy.simplify.radsimp import collect
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u, v = symbols('u,v', cls=Dummy)
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H = h.as_expr().subs({t: u + I*v}).expand()
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Q = q.as_expr().subs({t: u + I*v}).expand()
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H_map = collect(H, I, evaluate=False)
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Q_map = collect(Q, I, evaluate=False)
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a, b = H_map.get(S.One, S.Zero), H_map.get(I, S.Zero)
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c, d = Q_map.get(S.One, S.Zero), Q_map.get(I, S.Zero)
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R = Poly(resultant(c, d, v), u)
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R_u = roots(R, filter='R')
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if len(R_u) != R.count_roots():
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return None
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result = S.Zero
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for r_u in R_u.keys():
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|
|
C = Poly(c.subs({u: r_u}), v)
|
||
|
|
R_v = roots(C, filter='R')
|
||
|
|
|
||
|
|
if len(R_v) != C.count_roots():
|
||
|
|
return None
|
||
|
|
|
||
|
|
R_v_paired = [] # take one from each pair of conjugate roots
|
||
|
|
for r_v in R_v:
|
||
|
|
if r_v not in R_v_paired and -r_v not in R_v_paired:
|
||
|
|
if r_v.is_negative or r_v.could_extract_minus_sign():
|
||
|
|
R_v_paired.append(-r_v)
|
||
|
|
elif not r_v.is_zero:
|
||
|
|
R_v_paired.append(r_v)
|
||
|
|
|
||
|
|
for r_v in R_v_paired:
|
||
|
|
|
||
|
|
D = d.subs({u: r_u, v: r_v})
|
||
|
|
|
||
|
|
if D.evalf(chop=True) != 0:
|
||
|
|
continue
|
||
|
|
|
||
|
|
A = Poly(a.subs({u: r_u, v: r_v}), x)
|
||
|
|
B = Poly(b.subs({u: r_u, v: r_v}), x)
|
||
|
|
|
||
|
|
AB = (A**2 + B**2).as_expr()
|
||
|
|
|
||
|
|
result += r_u*log(AB) + r_v*log_to_atan(A, B)
|
||
|
|
|
||
|
|
R_q = roots(q, filter='R')
|
||
|
|
|
||
|
|
if len(R_q) != q.count_roots():
|
||
|
|
return None
|
||
|
|
|
||
|
|
for r in R_q.keys():
|
||
|
|
result += r*log(h.as_expr().subs(t, r))
|
||
|
|
|
||
|
|
return result
|