def _expr_big(cls, z, n): if n.is_even: return (n - Rational(1, 2)) * pi * I + log( sqrt(z) / 2) + I * asin(1 / sqrt(z)) else: return (n - Rational(1, 2)) * pi * I + log( sqrt(z) / 2) - I * asin(1 / sqrt(z))
def test_trigintegrate_mixed(): assert trigintegrate(sin(x) * sec(x), x) == -log(sin(x)**2 - 1) / 2 assert trigintegrate(sin(x) * csc(x), x) == x assert trigintegrate(sin(x) * cot(x), x) == sin(x) assert trigintegrate(cos(x) * sec(x), x) == x assert trigintegrate(cos(x) * csc(x), x) == log(cos(x)**2 - 1) / 2 assert trigintegrate(cos(x) * tan(x), x) == -cos(x) assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \ - log(cos(x) + 1)/2 + cos(x)
def test_trigintegrate_mixed(): assert trigintegrate(sin(x)*sec(x), x) == -log(sin(x)**2 - 1)/2 assert trigintegrate(sin(x)*csc(x), x) == x assert trigintegrate(sin(x)*cot(x), x) == sin(x) assert trigintegrate(cos(x)*sec(x), x) == x assert trigintegrate(cos(x)*csc(x), x) == log(cos(x)**2 - 1)/2 assert trigintegrate(cos(x)*tan(x), x) == -cos(x) assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \ - log(cos(x) + 1)/2 + cos(x)
def test_catalan(): n = Symbol('n', integer=True) m = Symbol('n', integer=True, positive=True) catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786] for i, c in enumerate(catalans): assert catalan(i) == c assert catalan(n).rewrite(factorial).subs(n, i) == c assert catalan(n).rewrite(Product).subs(n, i).doit() == c assert catalan(x) == catalan(x) assert catalan(2 * x).rewrite(binomial) == binomial(4 * x, 2 * x) / (2 * x + 1) assert catalan(Rational(1, 2)).rewrite(gamma) == 8 / (3 * pi) assert catalan(Rational(1, 2)).rewrite(factorial).rewrite(gamma) ==\ 8 / (3 * pi) assert catalan(3 * x).rewrite(gamma) == 4**( 3 * x) * gamma(3 * x + Rational(1, 2)) / (sqrt(pi) * gamma(3 * x + 2)) assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2, ), 1) assert catalan(n).rewrite(factorial) == factorial( 2 * n) / (factorial(n + 1) * factorial(n)) assert isinstance(catalan(n).rewrite(Product), catalan) assert isinstance(catalan(m).rewrite(Product), Product) assert diff(catalan(x), x) == (polygamma(0, x + Rational(1, 2)) - polygamma(0, x + 2) + log(4)) * catalan(x) assert catalan(x).evalf() == catalan(x) c = catalan(S.Half).evalf() assert str(c) == '0.848826363156775' c = catalan(I).evalf(3) assert sstr((re(c), im(c))) == '(0.398, -0.0209)'
def test_catalan(): n = Symbol('n', integer=True) m = Symbol('n', integer=True, positive=True) catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786] for i, c in enumerate(catalans): assert catalan(i) == c assert catalan(n).rewrite(factorial).subs({n: i}) == c assert catalan(n).rewrite(Product).subs({n: i}).doit() == c assert catalan(x) == catalan(x) assert catalan(2*x).rewrite(binomial) == binomial(4*x, 2*x)/(2*x + 1) assert catalan(Rational(1, 2)).rewrite(gamma) == 8/(3*pi) assert catalan(Rational(1, 2)).rewrite(factorial).rewrite(gamma) ==\ 8 / (3 * pi) assert catalan(3*x).rewrite(gamma) == 4**( 3*x)*gamma(3*x + Rational(1, 2))/(sqrt(pi)*gamma(3*x + 2)) assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2,), 1) assert catalan(n).rewrite(factorial) == factorial(2*n) / (factorial(n + 1) * factorial(n)) assert isinstance(catalan(n).rewrite(Product), catalan) assert isinstance(catalan(m).rewrite(Product), Product) assert diff(catalan(x), x) == (polygamma( 0, x + Rational(1, 2)) - polygamma(0, x + 2) + log(4))*catalan(x) assert catalan(x).evalf() == catalan(x) c = catalan(Rational(1, 2)).evalf() assert str(c) == '0.848826363156775' c = catalan(I).evalf(3) assert sstr((re(c), im(c))) == '(0.398, -0.0209)'
def singularities(f, x): """Find singularities of real-valued function `f` with respect to `x`. Examples ======== >>> from diofant import Symbol, exp, log >>> from diofant.abc import x >>> singularities(1/(1 + x), x) == {-1} True >>> singularities(exp(1/x) + log(x + 1), x) == {-1, 0} True >>> singularities(exp(1/log(x + 1)), x) == {0} True Notes ===== Removable singularities are not supported now. References ========== .. [1] http://en.wikipedia.org/wiki/Mathematical_singularity """ f, x = sympify(f), sympify(x) guess, res = set(), set() assert x.is_Symbol if f.is_number: return set() elif f.is_polynomial(x): return set() elif f.func in (Add, Mul): guess = guess.union(*[singularities(a, x) for a in f.args]) elif f.func is Pow: if f.exp.is_number and f.exp.is_negative: guess = {v for v in solve(f.base, x) if v.is_real} else: guess |= singularities(log(f.base)*f.exp, x) elif f.func in (log, sign) and len(f.args) == 1: guess |= singularities(f.args[0], x) guess |= {v for v in solve(f.args[0], x) if v.is_real} else: # pragma: no cover raise NotImplementedError for s in guess: l = Limit(f, x, s, dir="real") try: r = l.doit() if r == l or f.subs(x, s) != r: # pragma: no cover raise NotImplementedError except PoleError: res.add(s) return res
def test_Function(): assert mcode(f(x, y, z)) == "f[x, y, z]" assert mcode(sin(x)**cos(x)) == "Sin[x]^Cos[x]" assert mcode(sign(x)) == "Sign[x]" assert mcode(atanh(x), user_functions={"atanh": "ArcTanh"}) == "ArcTanh[x]" assert (mcode(meijerg(((1, 1), (3, 4)), ((1, ), ()), x)) == "MeijerG[{{1, 1}, {3, 4}}, {{1}, {}}, x]") assert (mcode(hyper((1, 2, 3), (3, 4), x)) == "HypergeometricPFQ[{1, 2, 3}, {3, 4}, x]") assert mcode(Min(x, y)) == "Min[x, y]" assert mcode(Max(x, y)) == "Max[x, y]" assert mcode(Max(x, 2)) == "Max[2, x]" # issue sympy/sympy#15344 assert mcode(binomial(x, y)) == "Binomial[x, y]" assert mcode(log(x)) == "Log[x]" assert mcode(tan(x)) == "Tan[x]" assert mcode(cot(x)) == "Cot[x]" assert mcode(asin(x)) == "ArcSin[x]" assert mcode(acos(x)) == "ArcCos[x]" assert mcode(atan(x)) == "ArcTan[x]" assert mcode(sinh(x)) == "Sinh[x]" assert mcode(cosh(x)) == "Cosh[x]" assert mcode(tanh(x)) == "Tanh[x]" assert mcode(coth(x)) == "Coth[x]" assert mcode(sech(x)) == "Sech[x]" assert mcode(csch(x)) == "Csch[x]" assert mcode(erfc(x)) == "Erfc[x]" assert mcode(conjugate(x)) == "Conjugate[x]" assert mcode(re(x)) == "Re[x]" assert mcode(im(x)) == "Im[x]" assert mcode(polygamma(x, y)) == "PolyGamma[x, y]" class myfunc1(Function): @classmethod def eval(cls, x): pass class myfunc2(Function): @classmethod def eval(cls, x, y): pass pytest.raises( ValueError, lambda: mcode(myfunc1(x), user_functions={"myfunc1": ["Myfunc1"]})) assert mcode(myfunc1(x), user_functions={"myfunc1": "Myfunc1"}) == "Myfunc1[x]" assert mcode(myfunc2(x, y), user_functions={"myfunc2": [(lambda *x: False, "Myfunc2")] }) == "myfunc2[x, y]"
def compare(a, b, x): r""" Determine order relation between two functons. Returns ======= {1, 0, -1} Respectively, if `a(x) \succ b(x)`, `a(x) \asymp b(x)` or `b(x) \succ a(x)`. Examples ======== >>> from diofant import Symbol, exp >>> x = Symbol('x', real=True, positive=True) >>> m = Symbol('m', real=True, positive=True) >>> compare(x, x**2, x) 0 >>> compare(1/x, x**m, x) 0 >>> compare(exp(x), exp(x**2), x) -1 >>> compare(exp(x), x**5, x) 1 """ # The log(exp(...)) must always be simplified here for termination. la = a.exp if a.is_Pow and a.base is S.Exp1 else log(a) lb = b.exp if b.is_Pow and b.base is S.Exp1 else log(b) c = limitinf(la / lb, x) if c.is_zero: return -1 elif c.is_infinite: return 1 else: return 0
def ispow2(d, log2=False): if not d.is_Pow: return False e = d.exp if e.is_Rational and e.q == 2 or symbolic and fraction(e)[1] == 2: return True if log2: q = 1 if e.is_Rational: q = e.q elif symbolic: d = fraction(e)[1] if d.is_Integer: q = d if q != 1 and log(q, 2).is_Integer: return True return False
def mrv_leadterm(e, x): """ Compute the leading term of the series. Returns ======= tuple The leading term `c_0 w^{e_0}` of the series of `e` in terms of the most rapidly varying subexpression `w` in form of the pair ``(c0, e0)`` of Expr. Examples ======== >>> from diofant import Symbol, exp >>> x = Symbol('x', real=True, positive=True) >>> mrv_leadterm(1/exp(-x + exp(-x)) - exp(x), x) (-1, 0) """ if not e.has(x): return e, S.Zero e = e.replace(lambda f: f.is_Pow and f.base != S.Exp1 and f.exp.has(x), lambda f: exp(log(f.base) * f.exp)) e = e.replace( lambda f: f.is_Mul and sum(a.is_Pow for a in f.args) > 1, lambda f: Mul( exp(Add(*[a.exp for a in f.args if a.is_Pow and a.base is S.Exp1])), * [a for a in f.args if not a.is_Pow or a.base is not S.Exp1])) # The positive dummy, w, is used here so log(w*2) etc. will expand. # TODO: For limits of complex functions, the algorithm would have to # be improved, or just find limits of Re and Im components separately. w = Dummy("w", real=True, positive=True) e, logw = rewrite(e, x, w) lt = e.compute_leading_term(w, logx=logw) return lt.as_coeff_exponent(w)
def _expr_big_minus(cls, x, n): return log(1 + x) + 2 * n * pi * I
def _expr_big(cls, x, n): return log(x - 1) + (2 * n - 1) * pi * I
def _expr_small_minus(cls, x): return log(1 + x)
def _expr_small(cls, x): return log(1 - x)
def test_harmonic_rational(): ne = Integer(6) no = Integer(5) pe = Integer(8) po = Integer(9) qe = Integer(10) qo = Integer(13) Heee = harmonic(ne + pe / qe) Aeee = (-log(10) + 2 * (-1 / Integer(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / Integer(8))) + 2 * (-sqrt(5) / 4 - 1 / Integer(4)) * log(sqrt(sqrt(5) / 8 + 5 / Integer(8))) + pi * (1 / Integer(4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + 5 / Integer(8))) + 13944145 / Integer(4720968)) Heeo = harmonic(ne + pe / qo) Aeeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + 2422020029 / Integer(702257080)) Heoe = harmonic(ne + po / qe) Aeoe = ( -log(20) + 2 * (1 / Integer(4) + sqrt(5) / 4) * log(-1 / Integer(4) + sqrt(5) / 4) + 2 * (-1 / Integer(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / Integer(8))) + 2 * (-sqrt(5) / 4 - 1 / Integer(4)) * log(sqrt(sqrt(5) / 8 + 5 / Integer(8))) + 2 * (-sqrt(5) / 4 + 1 / Integer(4)) * log(1 / Integer(4) + sqrt(5) / 4) + 11818877030 / Integer(4286604231) + pi * (sqrt(5) / 8 + 5 / Integer(8)) / sqrt(-sqrt(5) / 8 + 5 / Integer(8))) Heoo = harmonic(ne + po / qo) Aeoo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + 11669332571 / Integer(3628714320)) Hoee = harmonic(no + pe / qe) Aoee = (-log(10) + 2 * (-1 / Integer(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / Integer(8))) + 2 * (-sqrt(5) / 4 - 1 / Integer(4)) * log(sqrt(sqrt(5) / 8 + 5 / Integer(8))) + pi * (1 / Integer(4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + 5 / Integer(8))) + 779405 / Integer(277704)) Hoeo = harmonic(no + pe / qo) Aoeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + 53857323 / Integer(16331560)) Hooe = harmonic(no + po / qe) Aooe = ( -log(20) + 2 * (1 / Integer(4) + sqrt(5) / 4) * log(-1 / Integer(4) + sqrt(5) / 4) + 2 * (-1 / Integer(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / Integer(8))) + 2 * (-sqrt(5) / 4 - 1 / Integer(4)) * log(sqrt(sqrt(5) / 8 + 5 / Integer(8))) + 2 * (-sqrt(5) / 4 + 1 / Integer(4)) * log(1 / Integer(4) + sqrt(5) / 4) + 486853480 / Integer(186374097) + pi * (sqrt(5) / 8 + 5 / Integer(8)) / sqrt(-sqrt(5) / 8 + 5 / Integer(8))) Hooo = harmonic(no + po / qo) Aooo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + 383693479 / Integer(125128080)) H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo] A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo] for h, a in zip(H, A): e = expand_func(h).doit() assert cancel(e / a) == 1 assert h.n() == a.n()
def test_Function(): assert mcode(f(x, y, z)) == "f[x, y, z]" assert mcode(sin(x) ** cos(x)) == "Sin[x]^Cos[x]" assert mcode(sign(x)) == "Sign[x]" assert mcode(atanh(x), user_functions={"atanh": "ArcTanh"}) == "ArcTanh[x]" assert (mcode(meijerg(((1, 1), (3, 4)), ((1,), ()), x)) == "MeijerG[{{1, 1}, {3, 4}}, {{1}, {}}, x]") assert (mcode(hyper((1, 2, 3), (3, 4), x)) == "HypergeometricPFQ[{1, 2, 3}, {3, 4}, x]") assert mcode(Min(x, y)) == "Min[x, y]" assert mcode(Max(x, y)) == "Max[x, y]" assert mcode(Max(x, 2)) == "Max[2, x]" # issue sympy/sympy#15344 assert mcode(binomial(x, y)) == "Binomial[x, y]" assert mcode(log(x)) == "Log[x]" assert mcode(tan(x)) == "Tan[x]" assert mcode(cot(x)) == "Cot[x]" assert mcode(asin(x)) == "ArcSin[x]" assert mcode(acos(x)) == "ArcCos[x]" assert mcode(atan(x)) == "ArcTan[x]" assert mcode(sinh(x)) == "Sinh[x]" assert mcode(cosh(x)) == "Cosh[x]" assert mcode(tanh(x)) == "Tanh[x]" assert mcode(coth(x)) == "Coth[x]" assert mcode(sech(x)) == "Sech[x]" assert mcode(csch(x)) == "Csch[x]" assert mcode(erfc(x)) == "Erfc[x]" assert mcode(conjugate(x)) == "Conjugate[x]" assert mcode(re(x)) == "Re[x]" assert mcode(im(x)) == "Im[x]" assert mcode(polygamma(x, y)) == "PolyGamma[x, y]" assert mcode(factorial(x)) == "Factorial[x]" assert mcode(factorial2(x)) == "Factorial2[x]" assert mcode(rf(x, y)) == "Pochhammer[x, y]" assert mcode(gamma(x)) == "Gamma[x]" assert mcode(zeta(x)) == "Zeta[x]" assert mcode(asinh(x)) == "ArcSinh[x]" assert mcode(Heaviside(x)) == "UnitStep[x]" assert mcode(fibonacci(x)) == "Fibonacci[x]" assert mcode(polylog(x, y)) == "PolyLog[x, y]" assert mcode(atanh(x)) == "ArcTanh[x]" class myfunc1(Function): @classmethod def eval(cls, x): pass class myfunc2(Function): @classmethod def eval(cls, x, y): pass pytest.raises(ValueError, lambda: mcode(myfunc1(x), user_functions={"myfunc1": ["Myfunc1"]})) assert mcode(myfunc1(x), user_functions={"myfunc1": "Myfunc1"}) == "Myfunc1[x]" assert mcode(myfunc2(x, y), user_functions={"myfunc2": [(lambda *x: False, "Myfunc2")]}) == "myfunc2[x, y]"
def test_RootSum(): r = RootSum(x**3 + x + 3, Lambda(y, log(y*z))) assert mcode(r) == ("RootSum[Function[{x}, x^3 + x + 3], " "Function[{y}, Log[y*z]]]")
def test_harmonic_rational(): ne = Integer(6) no = Integer(5) pe = Integer(8) po = Integer(9) qe = Integer(10) qo = Integer(13) Heee = harmonic(ne + pe/qe) Aeee = (-log(10) + 2*(Rational(-1, 4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + Rational(5, 8))) + 2*(-sqrt(5)/4 - Rational(1, 4))*log(sqrt(sqrt(5)/8 + Rational(5, 8))) + pi*(Rational(1, 4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + Rational(5, 8))) + Rational(13944145, 4720968)) Heeo = harmonic(ne + pe/qo) Aeeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13) + 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13) - 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2 - 2*log(sin(pi/13))*cos(3*pi/13) + Rational(2422020029, 702257080)) Heoe = harmonic(ne + po/qe) Aeoe = (-log(20) + 2*(Rational(1, 4) + sqrt(5)/4)*log(Rational(-1, 4) + sqrt(5)/4) + 2*(Rational(-1, 4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + Rational(5, 8))) + 2*(-sqrt(5)/4 - Rational(1, 4))*log(sqrt(sqrt(5)/8 + Rational(5, 8))) + 2*(-sqrt(5)/4 + Rational(1, 4))*log(Rational(1, 4) + sqrt(5)/4) + Rational(11818877030, 4286604231) + pi*(sqrt(5)/8 + Rational(5, 8))/sqrt(-sqrt(5)/8 + Rational(5, 8))) Heoo = harmonic(ne + po/qo) Aeoo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13) + 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13) - 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2 - 2*log(sin(2*pi/13))*cos(3*pi/13) + Rational(11669332571, 3628714320)) Hoee = harmonic(no + pe/qe) Aoee = (-log(10) + 2*(Rational(-1, 4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + Rational(5, 8))) + 2*(-sqrt(5)/4 - Rational(1, 4))*log(sqrt(sqrt(5)/8 + Rational(5, 8))) + pi*(Rational(1, 4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + Rational(5, 8))) + Rational(779405, 277704)) Hoeo = harmonic(no + pe/qo) Aoeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13) + 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13) - 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2 - 2*log(sin(pi/13))*cos(3*pi/13) + Rational(53857323, 16331560)) Hooe = harmonic(no + po/qe) Aooe = (-log(20) + 2*(Rational(1, 4) + sqrt(5)/4)*log(Rational(-1, 4) + sqrt(5)/4) + 2*(Rational(-1, 4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + Rational(5, 8))) + 2*(-sqrt(5)/4 - Rational(1, 4))*log(sqrt(sqrt(5)/8 + Rational(5, 8))) + 2*(-sqrt(5)/4 + Rational(1, 4))*log(Rational(1, 4) + sqrt(5)/4) + Rational(486853480, 186374097) + pi*(sqrt(5)/8 + Rational(5, 8))/sqrt(-sqrt(5)/8 + Rational(5, 8))) Hooo = harmonic(no + po/qo) Aooo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13) + 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13) - 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2 - 2*log(sin(2*pi/13))*cos(3*pi/13) + Rational(383693479, 125128080)) H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo] A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo] for h, a in zip(H, A): e = expand_func(h).doit() assert cancel(e/a) == 1 assert h.evalf() == a.evalf()
def test_harmonic_rational(): ne = Integer(6) no = Integer(5) pe = Integer(8) po = Integer(9) qe = Integer(10) qo = Integer(13) Heee = harmonic(ne + pe / qe) Aeee = (-log(10) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + pi * (Rational(1, 4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + Rational(5, 8))) + Rational(13944145, 4720968)) Heeo = harmonic(ne + pe / qo) Aeeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + Rational(2422020029, 702257080)) Heoe = harmonic(ne + po / qe) Aeoe = ( -log(20) + 2 * (Rational(1, 4) + sqrt(5) / 4) * log(Rational(-1, 4) + sqrt(5) / 4) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 + Rational(1, 4)) * log(Rational(1, 4) + sqrt(5) / 4) + Rational(11818877030, 4286604231) + pi * (sqrt(5) / 8 + Rational(5, 8)) / sqrt(-sqrt(5) / 8 + Rational(5, 8))) Heoo = harmonic(ne + po / qo) Aeoo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + Rational(11669332571, 3628714320)) Hoee = harmonic(no + pe / qe) Aoee = (-log(10) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + pi * (Rational(1, 4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + Rational(5, 8))) + Rational(779405, 277704)) Hoeo = harmonic(no + pe / qo) Aoeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + Rational(53857323, 16331560)) Hooe = harmonic(no + po / qe) Aooe = ( -log(20) + 2 * (Rational(1, 4) + sqrt(5) / 4) * log(Rational(-1, 4) + sqrt(5) / 4) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 + Rational(1, 4)) * log(Rational(1, 4) + sqrt(5) / 4) + Rational(486853480, 186374097) + pi * (sqrt(5) / 8 + Rational(5, 8)) / sqrt(-sqrt(5) / 8 + Rational(5, 8))) Hooo = harmonic(no + po / qo) Aooo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + Rational(383693479, 125128080)) H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo] A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo] for h, a in zip(H, A): e = expand_func(h).doit() assert cancel(e / a) == 1 assert h.evalf() == a.evalf()
def _expr_small(cls, z): return log(Rational(1, 2) + sqrt(1 - z) / 2)
def _expr_small_minus(cls, z): return log(Rational(1, 2) + sqrt(1 + z) / 2)
def f(rv): if not (rv.is_Add or rv.is_Mul): return rv def gooda(a): # bool to tell whether the leading ``a`` in ``a*log(x)`` # could appear as log(x**a) return (a is not S.NegativeOne and # -1 *could* go, but we disallow (a.is_extended_real or force and a.is_extended_real is not False)) def goodlog(l): # bool to tell whether log ``l``'s argument can combine with others a = l.args[0] return a.is_positive or force and a.is_nonpositive is not False other = [] logs = [] log1 = defaultdict(list) for a in Add.make_args(rv): if a.func is log and goodlog(a): log1[()].append(([], a)) elif not a.is_Mul: other.append(a) else: ot = [] co = [] lo = [] for ai in a.args: if ai.is_Rational and ai < 0: ot.append(S.NegativeOne) co.append(-ai) elif ai.func is log and goodlog(ai): lo.append(ai) elif gooda(ai): co.append(ai) else: ot.append(ai) if len(lo) > 1: logs.append((ot, co, lo)) elif lo: log1[tuple(ot)].append((co, lo[0])) else: other.append(a) # if there is only one log at each coefficient and none have # an exponent to place inside the log then there is nothing to do if not logs and all( len(log1[k]) == 1 and log1[k][0] == [] for k in log1): return rv # collapse multi-logs as far as possible in a canonical way # TODO: see if x*log(a)+x*log(a)*log(b) -> x*log(a)*(1+log(b))? # -- in this case, it's unambiguous, but if it were were a log(c) in # each term then it's arbitrary whether they are grouped by log(a) or # by log(c). So for now, just leave this alone; it's probably better to # let the user decide for o, e, l in logs: l = list(ordered(l)) e = log(l.pop(0).args[0]**Mul(*e)) while l: li = l.pop(0) e = log(li.args[0]**e) c, l = Mul(*o), e if l.func is log: # it should be, but check to be sure log1[(c, )].append(([], l)) else: other.append(c * l) # logs that have the same coefficient can multiply for k in list(log1.keys()): log1[Mul(*k)] = log( logcombine(Mul(*[l.args[0]**Mul(*c) for c, l in log1.pop(k)]), force=force)) # logs that have oppositely signed coefficients can divide for k in ordered(list(log1.keys())): if k not in log1: # already popped as -k continue if -k in log1: # figure out which has the minus sign; the one with # more op counts should be the one num, den = k, -k if num.count_ops() > den.count_ops(): num, den = den, num other.append( num * log(log1.pop(num).args[0] / log1.pop(den).args[0])) else: other.append(k * log1.pop(k)) return Add(*other)
def _expr_big_minus(self, z, n): if n.is_even: return pi * I * n + log(Rational(1, 2) + sqrt(1 + z) / 2) else: return pi * I * n + log(sqrt(1 + z) / 2 - Rational(1, 2))
def _integrate(field=None): irreducibles = set() for poly in reducibles: for z in poly.free_symbols: if z in V: break # should this be: `irreducibles |= \ else: # set(root_factors(poly, z, filter=field))` continue # and the line below deleted? # | # V irreducibles |= set(root_factors(poly, z, filter=field)) log_coeffs, log_part = [], [] B = _symbols('B', len(irreducibles)) # Note: the ordering matters here for poly, b in reversed(list(ordered(zip(irreducibles, B)))): if poly.has(*V): poly_coeffs.append(b) log_part.append(b * log(poly)) # TODO: Currently it's better to use symbolic expressions here instead # of rational functions, because it's simpler and FracElement doesn't # give big speed improvement yet. This is because cancelation is slow # due to slow polynomial GCD algorithms. If this gets improved then # revise this code. candidate = poly_part / poly_denom + Add(*log_part) h = F - _derivation(candidate) / denom raw_numer = h.as_numer_denom()[0] # Rewrite raw_numer as a polynomial in K[coeffs][V] where K is a field # that we have to determine. We can't use simply atoms() because log(3), # sqrt(y) and similar expressions can appear, leading to non-trivial # domains. syms = set(poly_coeffs) | set(V) non_syms = set() def find_non_syms(expr): if expr.is_Integer or expr.is_Rational: pass # ignore trivial numbers elif expr in syms: pass # ignore variables elif not expr.has(*syms): non_syms.add(expr) elif expr.is_Add or expr.is_Mul or expr.is_Pow: list(map(find_non_syms, expr.args)) else: # TODO: Non-polynomial expression. This should have been # filtered out at an earlier stage. raise PolynomialError try: find_non_syms(raw_numer) except PolynomialError: return else: ground, _ = construct_domain(non_syms, field=True) coeff_ring = PolyRing(poly_coeffs, ground) ring = PolyRing(V, coeff_ring) numer = ring.from_expr(raw_numer) solution = solve_lin_sys(numer.coeffs(), coeff_ring) if solution is None: return else: solution = [(coeff_ring.symbols[coeff_ring.index(k)], v.as_expr()) for k, v in solution.items()] return candidate.subs(solution).subs( list(zip(poly_coeffs, [S.Zero] * len(poly_coeffs))))
def _denest_pow(eq): """ Denest powers. This is a helper function for powdenest that performs the actual transformation. """ from diofant.simplify.simplify import logcombine b, e = eq.as_base_exp() if b.is_Pow and e != 1: new = b._eval_power(e) if new is not None: eq = new b, e = new.as_base_exp() # denest exp with log terms in exponent if b is S.Exp1 and e.is_Mul: logs = [] other = [] for ei in e.args: if any(ai.func is log for ai in Add.make_args(ei)): logs.append(ei) else: other.append(ei) logs = logcombine(Mul(*logs)) return Pow(exp(logs), Mul(*other)) _, be = b.as_base_exp() if be is S.One and not (b.is_Mul or b.is_Rational and b.q != 1 or b.is_positive): return eq # denest eq which is either pos**e or Pow**e or Mul**e or # Mul(b1**e1, b2**e2) # handle polar numbers specially polars, nonpolars = [], [] for bb in Mul.make_args(b): if bb.is_polar: polars.append(bb.as_base_exp()) else: nonpolars.append(bb) if len(polars) == 1 and not polars[0][0].is_Mul: return Pow(polars[0][0], polars[0][1] * e) * powdenest( Mul(*nonpolars)**e) elif polars: return Mul(*[powdenest(bb**(ee*e)) for (bb, ee) in polars]) \ * powdenest(Mul(*nonpolars)**e) if b.is_Integer: # use log to see if there is a power here logb = expand_log(log(b)) if logb.is_Mul: c, logb = logb.args e *= c base = logb.args[0] return Pow(base, e) # if b is not a Mul or any factor is an atom then there is nothing to do if not b.is_Mul or any(s.is_Atom for s in Mul.make_args(b)): return eq # let log handle the case of the base of the argument being a Mul, e.g. # sqrt(x**(2*i)*y**(6*i)) -> x**i*y**(3**i) if x and y are positive; we # will take the log, expand it, and then factor out the common powers that # now appear as coefficient. We do this manually since terms_gcd pulls out # fractions, terms_gcd(x+x*y/2) -> x*(y + 2)/2 and we don't want the 1/2; # gcd won't pull out numerators from a fraction: gcd(3*x, 9*x/2) -> x but # we want 3*x. Neither work with noncommutatives. def nc_gcd(aa, bb): a, b = [i.as_coeff_Mul() for i in [aa, bb]] c = gcd(a[0], b[0]).as_numer_denom()[0] g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0])) return _keep_coeff(c, g) glogb = expand_log(log(b)) if glogb.is_Add: args = glogb.args g = reduce(nc_gcd, args) if g != 1: cg, rg = g.as_coeff_Mul() glogb = _keep_coeff(cg, rg * Add(*[a / g for a in args])) # now put the log back together again if glogb.func is log or not glogb.is_Mul: if glogb.args[0].is_Pow: glogb = _denest_pow(glogb.args[0]) if (abs(glogb.exp) < 1) is S.true: return Pow(glogb.base, glogb.exp * e) return eq # the log(b) was a Mul so join any adds with logcombine add = [] other = [] for a in glogb.args: if a.is_Add: add.append(a) else: other.append(a) return Pow(exp(logcombine(Mul(*add))), e * Mul(*other))
def _diff_wrt_parameter(self, idx): # Differentiation wrt a parameter can only be done in very special # cases. In particular, if we want to differentiate with respect to # `a`, all other gamma factors have to reduce to rational functions. # # Let MT denote mellin transform. Suppose T(-s) is the gamma factor # appearing in the definition of G. Then # # MT(log(z)G(z)) = d/ds T(s) = d/da T(s) + ... # # Thus d/da G(z) = log(z)G(z) - ... # The ... can be evaluated as a G function under the above conditions, # the formula being most easily derived by using # # d Gamma(s + n) Gamma(s + n) / 1 1 1 \ # -- ------------ = ------------ | - + ---- + ... + --------- | # ds Gamma(s) Gamma(s) \ s s + 1 s + n - 1 / # # which follows from the difference equation of the digamma function. # (There is a similar equation for -n instead of +n). # We first figure out how to pair the parameters. an = list(self.an) ap = list(self.aother) bm = list(self.bm) bq = list(self.bother) if idx < len(an): an.pop(idx) else: idx -= len(an) if idx < len(ap): ap.pop(idx) else: idx -= len(ap) if idx < len(bm): bm.pop(idx) else: bq.pop(idx - len(bm)) pairs1 = [] pairs2 = [] for l1, l2, pairs in [(an, bq, pairs1), (ap, bm, pairs2)]: while l1: x = l1.pop() found = None for i, y in enumerate(l2): if not Mod((x - y).simplify(), 1): found = i break if found is None: raise NotImplementedError('Derivative not expressible ' 'as G-function?') y = l2[i] l2.pop(i) pairs.append((x, y)) # Now build the result. res = log(self.argument) * self for a, b in pairs1: sign = 1 n = a - b base = b if n < 0: sign = -1 n = b - a base = a for k in range(n): res -= sign * meijerg(self.an + (base + k + 1, ), self.aother, self.bm, self.bother + (base + k + 0, ), self.argument) for a, b in pairs2: sign = 1 n = b - a base = a if n < 0: sign = -1 n = a - b base = b for k in range(n): res -= sign * meijerg( self.an, self.aother + (base + k + 1, ), self.bm + (base + k + 0, ), self.bother, self.argument) return res
def test_RootSum(): r = RootSum(x**3 + x + 3, Lambda(y, log(y*z))) assert mcode(r) == ("RootSum[Function[{x}, x^3 + x + 3], " "Function[{y}, Log[y*z]]]")
def heurisch(f, x, rewrite=False, hints=None, mappings=None, retries=3, degree_offset=0, unnecessary_permutations=None): """ Compute indefinite integral using heuristic Risch algorithm. This is a heuristic approach to indefinite integration in finite terms using the extended heuristic (parallel) Risch algorithm, based on Manuel Bronstein's "Poor Man's Integrator" [1]_. The algorithm supports various classes of functions including transcendental elementary or special functions like Airy, Bessel, Whittaker and Lambert. Note that this algorithm is not a decision procedure. If it isn't able to compute the antiderivative for a given function, then this is not a proof that such a functions does not exist. One should use recursive Risch algorithm in such case. It's an open question if this algorithm can be made a full decision procedure. This is an internal integrator procedure. You should use toplevel 'integrate' function in most cases, as this procedure needs some preprocessing steps and otherwise may fail. Parameters ========== heurisch(f, x, rewrite=False, hints=None) f : Expr expression x : Symbol variable rewrite : Boolean, optional force rewrite 'f' in terms of 'tan' and 'tanh', default False. hints : None or list a list of functions that may appear in anti-derivate. If None (default) - no suggestions at all, if empty list - try to figure out. Examples ======== >>> from diofant import tan >>> from diofant.integrals.heurisch import heurisch >>> from diofant.abc import x, y >>> heurisch(y*tan(x), x) y*log(tan(x)**2 + 1)/2 References ========== .. [1] Manuel Bronstein's "Poor Man's Integrator", http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html .. [2] K. Geddes, L. Stefanus, On the Risch-Norman Integration Method and its Implementation in Maple, Proceedings of ISSAC'89, ACM Press, 212-217. .. [3] J. H. Davenport, On the Parallel Risch Algorithm (I), Proceedings of EUROCAM'82, LNCS 144, Springer, 144-157. .. [4] J. H. Davenport, On the Parallel Risch Algorithm (III): Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6. .. [5] J. H. Davenport, B. M. Trager, On the Parallel Risch Algorithm (II), ACM Transactions on Mathematical Software 11 (1985), 356-362. See Also ======== diofant.integrals.integrals.Integral.doit diofant.integrals.integrals.Integral diofant.integrals.heurisch.components """ f = sympify(f) if x not in f.free_symbols: return f * x if not f.is_Add: indep, f = f.as_independent(x) else: indep = S.One rewritables = { (sin, cos, cot): tan, (sinh, cosh, coth): tanh, } if rewrite: for candidates, rule in rewritables.items(): f = f.rewrite(candidates, rule) else: for candidates in rewritables.keys(): if f.has(*candidates): break else: rewrite = True terms = components(f, x) if hints is not None: if not hints: a = Wild('a', exclude=[x]) b = Wild('b', exclude=[x]) c = Wild('c', exclude=[x]) for g in set(terms): # using copy of terms if g.is_Function: if g.func is li: M = g.args[0].match(a * x**b) if M is not None: terms.add( x * (li(M[a] * x**M[b]) - (M[a] * x**M[b])**(-1 / M[b]) * Ei( (M[b] + 1) * log(M[a] * x**M[b]) / M[b]))) # terms.add( x*(li(M[a]*x**M[b]) - (x**M[b])**(-1/M[b])*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) ) # terms.add( x*(li(M[a]*x**M[b]) - x*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) ) # terms.add( li(M[a]*x**M[b]) - Ei((M[b]+1)*log(M[a]*x**M[b])/M[b]) ) elif g.is_Pow: if g.base is S.Exp1: M = g.exp.match(a * x**2) if M is not None: if M[a].is_positive: terms.add(erfi(sqrt(M[a]) * x)) else: # M[a].is_negative or unknown terms.add(erf(sqrt(-M[a]) * x)) M = g.exp.match(a * x**2 + b * x + c) if M is not None: if M[a].is_positive: terms.add( sqrt(pi / 4 * (-M[a])) * exp(M[c] - M[b]**2 / (4 * M[a])) * erfi( sqrt(M[a]) * x + M[b] / (2 * sqrt(M[a])))) elif M[a].is_negative: terms.add( sqrt(pi / 4 * (-M[a])) * exp(M[c] - M[b]**2 / (4 * M[a])) * erf( sqrt(-M[a]) * x - M[b] / (2 * sqrt(-M[a])))) M = g.exp.match(a * log(x)**2) if M is not None: if M[a].is_positive: terms.add( erfi( sqrt(M[a]) * log(x) + 1 / (2 * sqrt(M[a])))) if M[a].is_negative: terms.add( erf( sqrt(-M[a]) * log(x) - 1 / (2 * sqrt(-M[a])))) elif g.exp.is_Rational and g.exp.q == 2: M = g.base.match(a * x**2 + b) if M is not None and M[b].is_positive: if M[a].is_positive: terms.add(asinh(sqrt(M[a] / M[b]) * x)) elif M[a].is_negative: terms.add(asin(sqrt(-M[a] / M[b]) * x)) M = g.base.match(a * x**2 - b) if M is not None and M[b].is_positive: if M[a].is_positive: terms.add(acosh(sqrt(M[a] / M[b]) * x)) elif M[a].is_negative: terms.add((-M[b] / 2 * sqrt(-M[a]) * atan( sqrt(-M[a]) * x / sqrt(M[a] * x**2 - M[b])) )) else: terms |= set(hints) for g in set(terms): # using copy of terms terms |= components(cancel(g.diff(x)), x) # TODO: caching is significant factor for why permutations work at all. Change this. V = _symbols('x', len(terms)) # sort mapping expressions from largest to smallest (last is always x). mapping = list( reversed( list( zip(*ordered( # [(a[0].as_independent(x)[1], a) for a in zip(terms, V)])))[1])) # rev_mapping = {v: k for k, v in mapping} # if mappings is None: # # optimizing the number of permutations of mapping # assert mapping[-1][0] == x # if not, find it and correct this comment unnecessary_permutations = [mapping.pop(-1)] mappings = permutations(mapping) else: unnecessary_permutations = unnecessary_permutations or [] def _substitute(expr): return expr.subs(mapping) for mapping in mappings: mapping = list(mapping) mapping = mapping + unnecessary_permutations diffs = [_substitute(cancel(g.diff(x))) for g in terms] denoms = [g.as_numer_denom()[1] for g in diffs] if all(h.is_polynomial(*V) for h in denoms) and _substitute(f).is_rational_function(*V): denom = reduce(lambda p, q: lcm(p, q, *V), denoms) break else: if not rewrite: result = heurisch( f, x, rewrite=True, hints=hints, unnecessary_permutations=unnecessary_permutations) if result is not None: return indep * result return numers = [cancel(denom * g) for g in diffs] def _derivation(h): return Add(*[d * h.diff(v) for d, v in zip(numers, V)]) def _deflation(p): for y in V: if not p.has(y): continue if _derivation(p) is not S.Zero: c, q = p.as_poly(y).primitive() return _deflation(c) * gcd(q, q.diff(y)).as_expr() else: return p def _splitter(p): for y in V: if not p.has(y): continue if _derivation(y) is not S.Zero: c, q = p.as_poly(y).primitive() q = q.as_expr() h = gcd(q, _derivation(q), y) s = quo(h, gcd(q, q.diff(y), y), y) c_split = _splitter(c) if s.as_poly(y).degree() == 0: return c_split[0], q * c_split[1] q_split = _splitter(cancel(q / s)) return c_split[0] * q_split[0] * s, c_split[1] * q_split[1] else: return S.One, p special = {} for term in terms: if term.is_Function: if term.func is tan: special[1 + _substitute(term)**2] = False elif term.func is tanh: special[1 + _substitute(term)] = False special[1 - _substitute(term)] = False elif term.func is LambertW: special[_substitute(term)] = True F = _substitute(f) P, Q = F.as_numer_denom() u_split = _splitter(denom) v_split = _splitter(Q) polys = set(list(v_split) + [u_split[0]] + list(special.keys())) s = u_split[0] * Mul(*[k for k, v in special.items() if v]) polified = [p.as_poly(*V) for p in [s, P, Q]] if None in polified: return # --- definitions for _integrate --- a, b, c = [p.total_degree() for p in polified] poly_denom = (s * v_split[0] * _deflation(v_split[1])).as_expr() def _exponent(g): if g.is_Pow: if g.exp.is_Rational and g.exp.q != 1: if g.exp.p > 0: return g.exp.p + g.exp.q - 1 else: return abs(g.exp.p + g.exp.q) else: return 1 elif not g.is_Atom and g.args: return max([_exponent(h) for h in g.args]) else: return 1 A, B = _exponent(f), a + max(b, c) if A > 1 and B > 1: monoms = itermonomials(V, A + B - 1 + degree_offset) else: monoms = itermonomials(V, A + B + degree_offset) poly_coeffs = _symbols('A', len(monoms)) poly_part = Add(*[ poly_coeffs[i] * monomial for i, monomial in enumerate(ordered(monoms)) ]) reducibles = set() for poly in polys: if poly.has(*V): try: factorization = factor(poly, greedy=True) except PolynomialError: factorization = poly factorization = poly if factorization.is_Mul: reducibles |= set(factorization.args) else: reducibles.add(factorization) def _integrate(field=None): irreducibles = set() for poly in reducibles: for z in poly.free_symbols: if z in V: break # should this be: `irreducibles |= \ else: # set(root_factors(poly, z, filter=field))` continue # and the line below deleted? # | # V irreducibles |= set(root_factors(poly, z, filter=field)) log_coeffs, log_part = [], [] B = _symbols('B', len(irreducibles)) # Note: the ordering matters here for poly, b in reversed(list(ordered(zip(irreducibles, B)))): if poly.has(*V): poly_coeffs.append(b) log_part.append(b * log(poly)) # TODO: Currently it's better to use symbolic expressions here instead # of rational functions, because it's simpler and FracElement doesn't # give big speed improvement yet. This is because cancelation is slow # due to slow polynomial GCD algorithms. If this gets improved then # revise this code. candidate = poly_part / poly_denom + Add(*log_part) h = F - _derivation(candidate) / denom raw_numer = h.as_numer_denom()[0] # Rewrite raw_numer as a polynomial in K[coeffs][V] where K is a field # that we have to determine. We can't use simply atoms() because log(3), # sqrt(y) and similar expressions can appear, leading to non-trivial # domains. syms = set(poly_coeffs) | set(V) non_syms = set() def find_non_syms(expr): if expr.is_Integer or expr.is_Rational: pass # ignore trivial numbers elif expr in syms: pass # ignore variables elif not expr.has(*syms): non_syms.add(expr) elif expr.is_Add or expr.is_Mul or expr.is_Pow: list(map(find_non_syms, expr.args)) else: # TODO: Non-polynomial expression. This should have been # filtered out at an earlier stage. raise PolynomialError try: find_non_syms(raw_numer) except PolynomialError: return else: ground, _ = construct_domain(non_syms, field=True) coeff_ring = PolyRing(poly_coeffs, ground) ring = PolyRing(V, coeff_ring) numer = ring.from_expr(raw_numer) solution = solve_lin_sys(numer.coeffs(), coeff_ring) if solution is None: return else: solution = [(coeff_ring.symbols[coeff_ring.index(k)], v.as_expr()) for k, v in solution.items()] return candidate.subs(solution).subs( list(zip(poly_coeffs, [S.Zero] * len(poly_coeffs)))) if not (F.free_symbols - set(V)): solution = _integrate('Q') if solution is None: solution = _integrate() else: solution = _integrate() if solution is not None: antideriv = solution.subs(rev_mapping) antideriv = cancel(antideriv).expand(force=True) if antideriv.is_Add: antideriv = antideriv.as_independent(x)[1] return indep * antideriv else: if retries >= 0: result = heurisch( f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations) if result is not None: return indep * result return