Python sympy.unpolarify函数代码示例

    def eval(cls, nu, z):
        if z.is_zero:
            if nu.is_zero:
                return S.One
            elif (nu.is_integer and nu.is_zero is False) or re(nu).is_positive:
                return S.Zero
            elif re(nu).is_negative and not (nu.is_integer is True):
                return S.ComplexInfinity
            elif nu.is_imaginary:
                return S.NaN
        if z is S.Infinity or (z is S.NegativeInfinity):
            return S.Zero

        if z.could_extract_minus_sign():
            return (z)**nu*(-z)**(-nu)*besselj(nu, -z)
        if nu.is_integer:
            if nu.could_extract_minus_sign():
                return S(-1)**(-nu)*besselj(-nu, z)
            newz = z.extract_multiplicatively(I)
            if newz:  # NOTE we don't want to change the function if z==0
                return I**(nu)*besseli(nu, newz)

        # branch handling:
        from sympy import unpolarify, exp
        if nu.is_integer:
            newz = unpolarify(z)
            if newz != z:
                return besselj(nu, newz)
        else:
            newz, n = z.extract_branch_factor()
            if n != 0:
                return exp(2*n*pi*nu*I)*besselj(nu, newz)
        nnu = unpolarify(nu)
        if nu != nnu:
            return besselj(nnu, z)

    def _eval_expand_log(self, deep=True, **hints):
        from sympy import unpolarify
        force = hints.get('force', False)
        arg = self.args[0]
        if arg.is_Mul:
            expr = []
            nonpos = []
            for x in arg.args:
                if force or x.is_positive or x.is_polar:
                    a = self.func(x)
                    if isinstance(a, log):
                        expr.append(self.func(x)._eval_expand_log(**hints))
                    else:
                        expr.append(a)
                else:
                    nonpos.append(x)
            return Add(*expr) + log(Mul(*nonpos))
        elif arg.is_Pow:
            if force or (arg.exp.is_real and arg.base.is_positive) or \
                    arg.base.is_polar:
                b = arg.base
                e = arg.exp
                a = self.func(b)
                if isinstance(a, log):
                    return unpolarify(e) * a._eval_expand_log(**hints)
                else:
                    return unpolarify(e) * a

        return self.func(arg)

def test_expint():
    """ Test various exponential integrals. """
    from sympy import (expint, unpolarify, Symbol, Ci, Si, Shi, Chi,
                       sin, cos, sinh, cosh, Ei)
    assert simplify(unpolarify(integrate(exp(-z*x)/x**y, (x, 1, oo),
                meijerg=True, conds='none'
                ).rewrite(expint).expand(func=True))) == expint(y, z)

    assert integrate(exp(-z*x)/x, (x, 1, oo), meijerg=True,
                     conds='none').rewrite(expint).expand() == \
        expint(1, z)
    assert integrate(exp(-z*x)/x**2, (x, 1, oo), meijerg=True,
                     conds='none').rewrite(expint).expand() == \
        expint(2, z).rewrite(Ei).rewrite(expint)
    assert integrate(exp(-z*x)/x**3, (x, 1, oo), meijerg=True,
                     conds='none').rewrite(expint).expand() == \
        expint(3, z).rewrite(Ei).rewrite(expint).expand()

    t = Symbol('t', positive=True)
    assert integrate(-cos(x)/x, (x, t, oo), meijerg=True).expand() == Ci(t)
    assert integrate(-sin(x)/x, (x, t, oo), meijerg=True).expand() == \
        Si(t) - pi/2
    assert integrate(sin(x)/x, (x, 0, z), meijerg=True) == Si(z)
    assert integrate(sinh(x)/x, (x, 0, z), meijerg=True) == Shi(z)
    assert integrate(exp(-x)/x, x, meijerg=True).expand().rewrite(expint) == \
        I*pi - expint(1, x)
    assert integrate(exp(-x)/x**2, x, meijerg=True).rewrite(expint).expand() \
        == expint(1, x) - exp(-x)/x - I*pi

    u = Symbol('u', polar=True)
    assert integrate(cos(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
        == Ci(u)
    assert integrate(cosh(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
        == Chi(u)

    assert integrate(expint(1, x), x, meijerg=True
            ).rewrite(expint).expand() == x*expint(1, x) - exp(-x)
    assert integrate(expint(2, x), x, meijerg=True
            ).rewrite(expint).expand() == \
        -x**2*expint(1, x)/2 + x*exp(-x)/2 - exp(-x)/2
    assert simplify(unpolarify(integrate(expint(y, x), x,
                 meijerg=True).rewrite(expint).expand(func=True))) == \
        -expint(y + 1, x)

    assert integrate(Si(x), x, meijerg=True) == x*Si(x) + cos(x)
    assert integrate(Ci(u), u, meijerg=True).expand() == u*Ci(u) - sin(u)
    assert integrate(Shi(x), x, meijerg=True) == x*Shi(x) - cosh(x)
    assert integrate(Chi(u), u, meijerg=True).expand() == u*Chi(u) - sinh(u)

    assert integrate(Si(x)*exp(-x), (x, 0, oo), meijerg=True) == pi/4
    assert integrate(expint(1, x)*sin(x), (x, 0, oo), meijerg=True) == log(2)/2

def can_do_meijer(a1, a2, b1, b2, numeric=True):
    """
    This helper function tries to hyperexpand() the meijer g-function
    corresponding to the parameters a1, a2, b1, b2.
    It returns False if this expansion still contains g-functions.
    If numeric is True, it also tests the so-obtained formula numerically
    (at random values) and returns False if the test fails.
    Else it returns True.
    """
    from sympy import unpolarify, expand
    r = hyperexpand(meijerg(a1, a2, b1, b2, z))
    if r.has(meijerg):
        return False
    # NOTE hyperexpand() returns a truly branched function, whereas numerical
    #      evaluation only works on the main branch. Since we are evaluating on
    #      the main branch, this should not be a problem, but expressions like
    #      exp_polar(I*pi/2*x)**a are evaluated incorrectly. We thus have to get
    #      rid of them. The expand heuristically does this...
    r = unpolarify(expand(r, force=True, power_base=True, power_exp=False,
                          mul=False, log=False, multinomial=False, basic=False))

    if not numeric:
        return True

    repl = {}
    for n, a in enumerate(meijerg(a1, a2, b1, b2, z).free_symbols - set([z])):
        repl[a] = randcplx(n)
    return tn(meijerg(a1, a2, b1, b2, z).subs(repl), r.subs(repl), z)

    def eval(cls, arg, base=None):
        from sympy import unpolarify
        if base is not None:
            base = sympify(base)

            if arg.is_positive and arg.is_Integer and \
               base.is_positive and base.is_Integer:
                base = int(base)
                arg = int(arg)
                n = multiplicity(base, arg)
                return S(n) + log(arg // base ** n) / log(base)
            if base is not S.Exp1:
                return cls(arg)/cls(base)
            else:
                return cls(arg)

        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.Zero:
                return S.ComplexInfinity
            elif arg is S.One:
                return S.Zero
            elif arg is S.Infinity:
                return S.Infinity
            elif arg is S.NegativeInfinity:
                return S.Infinity
            elif arg is S.NaN:
                return S.NaN
            elif arg.is_negative:
                return S.Pi * S.ImaginaryUnit + cls(-arg)
            elif arg.is_Rational:
                if arg.q != 1:
                    return cls(arg.p) - cls(arg.q)
                # remove perfect powers automatically
                p = perfect_power(int(arg))
                if p is not False:
                    return p[1]*cls(p[0])
        elif arg is S.ComplexInfinity:
            return S.ComplexInfinity
        elif arg is S.Exp1:
            return S.One
        elif arg.func is exp and arg.args[0].is_real:
            return arg.args[0]
        elif arg.func is exp_polar:
            return unpolarify(arg.exp)
        #don't autoexpand Pow or Mul (see the issue 252):
        elif not arg.is_Add:
            coeff = arg.as_coefficient(S.ImaginaryUnit)

            if coeff is not None:
                if coeff is S.Infinity:
                    return S.Infinity
                elif coeff is S.NegativeInfinity:
                    return S.Infinity
                elif coeff.is_Rational:
                    if coeff.is_nonnegative:
                        return S.Pi * S.ImaginaryUnit * S.Half + cls(coeff)
                    else:
                        return -S.Pi * S.ImaginaryUnit * S.Half + cls(-coeff)

 def eval(cls, s, z):
     s, z = sympify((s, z))
     if z == 1:
         return zeta(s)
     elif z == -1:
         return -dirichlet_eta(s)
     elif z == 0:
         return S.Zero
     elif s == 2:
         if z == S.Half:
             return pi**2/12 - log(2)**2/2
         elif z == 2:
             return pi**2/4 - I*pi*log(2)
         elif z == -(sqrt(5) - 1)/2:
             return -pi**2/15 + log((sqrt(5)-1)/2)**2/2
         elif z == -(sqrt(5) + 1)/2:
             return -pi**2/10 - log((sqrt(5)+1)/2)**2
         elif z == (3 - sqrt(5))/2:
             return pi**2/15 - log((sqrt(5)-1)/2)**2
         elif z == (sqrt(5) - 1)/2:
             return pi**2/10 - log((sqrt(5)-1)/2)**2
     # For s = 0 or -1 use explicit formulas to evaluate, but
     # automatically expanding polylog(1, z) to -log(1-z) seems undesirable
     # for summation methods based on hypergeometric functions
     elif s == 0:
         return z/(1 - z)
     elif s == -1:
         return z/(1 - z)**2
     # polylog is branched, but not over the unit disk
     from sympy.functions.elementary.complexes import (Abs, unpolarify,
         polar_lift)
     if z.has(exp_polar, polar_lift) and (Abs(z) <= S.One) == True:
         return cls(s, unpolarify(z))

    def _eval_expand_log(self, deep=True, **hints):
        from sympy import unpolarify
        force = hints.get('force', False)
        if deep:
            arg = self.args[0].expand(deep=deep, **hints)
        else:
            arg = self.args[0]
        if arg.is_Mul:
            expr = []
            nonpos = []
            for x in arg.args:
                if deep:
                    x = x.expand(deep=deep, **hints)
                if force or x.is_positive or x.is_polar:
                    expr.append(self.func(x)._eval_expand_log(deep=deep, **hints))
                else:
                    nonpos.append(x)
            return Add(*expr) + log(Mul(*nonpos))
        elif arg.is_Pow:
            if force or (arg.exp.is_real and arg.base.is_positive) or \
                        arg.base.is_polar:
                if deep:
                    b = arg.base.expand(deep=deep, **hints)
                    e = arg.exp.expand(deep=deep, **hints)
                else:
                    b = arg.base
                    e = arg.exp
                return unpolarify(e) * self.func(b)._eval_expand_log(deep=deep,\
                **hints)

        return self.func(arg)

    def _eval_expand_func(self, **hints):
        from sympy import exp, I, floor, Add, Poly, Dummy, exp_polar, unpolarify
        z, s, a = self.args
        if z == 1:
            return zeta(s, a)
        if s.is_Integer and s <= 0:
            t = Dummy('t')
            p = Poly((t + a)**(-s), t)
            start = 1/(1 - t)
            res = S(0)
            for c in reversed(p.all_coeffs()):
                res += c*start
                start = t*start.diff(t)
            return res.subs(t, z)

        if a.is_Rational:
            # See section 18 of
            #   Kelly B. Roach.  Hypergeometric Function Representations.
            #   In: Proceedings of the 1997 International Symposium on Symbolic and
            #   Algebraic Computation, pages 205-211, New York, 1997. ACM.
            # TODO should something be polarified here?
            add = S(0)
            mul = S(1)
            # First reduce a to the interaval (0, 1]
            if a > 1:
                n = floor(a)
                if n == a:
                    n -= 1
                a -= n
                mul = z**(-n)
                add = Add(*[-z**(k - n)/(a + k)**s for k in xrange(n)])
            elif a <= 0:
                n = floor(-a) + 1
                a += n
                mul = z**n
                add = Add(*[z**(n - 1 - k)/(a - k - 1)**s for k in xrange(n)])

            m, n = S([a.p, a.q])
            zet = exp_polar(2*pi*I/n)
            root = z**(1/n)
            return add + mul*n**(s - 1)*Add(
                *[polylog(s, zet**k*root)._eval_expand_func(**hints)
                  / (unpolarify(zet)**k*root)**m for k in xrange(n)])

        # TODO use minpoly instead of ad-hoc methods when issue 2789 is fixed
        if z.func is exp and (z.args[0]/(pi*I)).is_Rational or z in [-1, I, -I]:
            # TODO reference?
            if z == -1:
                p, q = S([1, 2])
            elif z == I:
                p, q = S([1, 4])
            elif z == -I:
                p, q = S([-1, 4])
            else:
                arg = z.args[0]/(2*pi*I)
                p, q = S([arg.p, arg.q])
            return Add(*[exp(2*pi*I*k*p/q)/q**s*zeta(s, (k + a)/q)
                         for k in xrange(q)])

        return lerchphi(z, s, a)

 def fdiff(self, argindex=1):
     from sympy import unpolarify
     arg = unpolarify(self.args[0])
     if argindex == 1:
         return C.exp(arg)/arg
     else:
         raise ArgumentIndexError(self, argindex)

    def eval(cls, n, z):
        n, z = list(map(sympify, (n, z)))
        from sympy import unpolarify

        if n.is_integer:
            if n.is_nonnegative:
                nz = unpolarify(z)
                if z != nz:
                    return polygamma(n, nz)

            if n == -1:
                return loggamma(z)
            else:
                if z.is_Number:
                    if z is S.NaN:
                        return S.NaN
                    elif z is S.Infinity:
                        if n.is_Number:
                            if n is S.Zero:
                                return S.Infinity
                            else:
                                return S.Zero
                    elif z.is_Integer:
                        if z.is_nonpositive:
                            return S.ComplexInfinity
                        else:
                            if n is S.Zero:
                                return -S.EulerGamma + C.harmonic(z - 1, 1)
                            elif n.is_odd:
                                return (-1) ** (n + 1) * C.factorial(n) * zeta(n + 1, z)

        if n == 0:
            if z is S.NaN:
                return S.NaN
            elif z.is_Rational:
                # TODO actually *any* n/m can be done, but that is messy
                lookup = {
                    S(1) / 2: -2 * log(2) - S.EulerGamma,
                    S(1) / 3: -S.Pi / 2 / sqrt(3) - 3 * log(3) / 2 - S.EulerGamma,
                    S(1) / 4: -S.Pi / 2 - 3 * log(2) - S.EulerGamma,
                    S(3) / 4: -3 * log(2) - S.EulerGamma + S.Pi / 2,
                    S(2) / 3: -3 * log(3) / 2 + S.Pi / 2 / sqrt(3) - S.EulerGamma,
                }
                if z > 0:
                    n = floor(z)
                    z0 = z - n
                    if z0 in lookup:
                        return lookup[z0] + Add(*[1 / (z0 + k) for k in range(n)])
                elif z < 0:
                    n = floor(1 - z)
                    z0 = z + n
                    if z0 in lookup:
                        return lookup[z0] - Add(*[1 / (z0 - 1 - k) for k in range(n)])
            elif z in (S.Infinity, S.NegativeInfinity):
                return S.Infinity
            else:
                t = z.extract_multiplicatively(S.ImaginaryUnit)
                if t in (S.Infinity, S.NegativeInfinity):
                    return S.Infinity

    def _eval_expand_log(self, deep=True, **hints):
        from sympy import unpolarify, expand_log
        from sympy.concrete import Sum, Product
        force = hints.get('force', False)
        if (len(self.args) == 2):
            return expand_log(self.func(*self.args), deep=deep, force=force)
        arg = self.args[0]
        if arg.is_Integer:
            # remove perfect powers
            p = perfect_power(int(arg))
            if p is not False:
                return p[1]*self.func(p[0])
        elif arg.is_Rational:
            return log(arg.p) - log(arg.q)
        elif arg.is_Mul:
            expr = []
            nonpos = []
            for x in arg.args:
                if force or x.is_positive or x.is_polar:
                    a = self.func(x)
                    if isinstance(a, log):
                        expr.append(self.func(x)._eval_expand_log(**hints))
                    else:
                        expr.append(a)
                elif x.is_negative:
                    a = self.func(-x)
                    expr.append(a)
                    nonpos.append(S.NegativeOne)
                else:
                    nonpos.append(x)
            return Add(*expr) + log(Mul(*nonpos))
        elif arg.is_Pow or isinstance(arg, exp):
            if force or (arg.exp.is_real and (arg.base.is_positive or ((arg.exp+1)
                .is_positive and (arg.exp-1).is_nonpositive))) or arg.base.is_polar:
                b = arg.base
                e = arg.exp
                a = self.func(b)
                if isinstance(a, log):
                    return unpolarify(e) * a._eval_expand_log(**hints)
                else:
                    return unpolarify(e) * a
        elif isinstance(arg, Product):
            if arg.function.is_positive:
                return Sum(log(arg.function), *arg.limits)

        return self.func(arg)

    def eval(cls, nu, z):
        if nu.is_Integer:
            newz = z.extract_multiplicatively(I)
            if newz:  # NOTE we don't want to change the function if z==0
                return I**(-nu)*besselj(nu, -newz)

        # branch handling:
        from sympy import unpolarify, exp
        if nu.is_integer:
            newz = unpolarify(z)
            if newz != z:
                return besseli(nu, newz)
        else:
            newz, n = z.extract_branch_factor()
            if n != 0:
                return exp(2*n*pi*nu*I)*besseli(nu, newz)
        nnu = unpolarify(nu)
        if nu != nnu:
            return besseli(nnu, z)

 def fdiff(self, argindex=2):
     from sympy import meijerg, unpolarify
     if argindex == 2:
         a, z = self.args
         return -C.exp(-unpolarify(z))*z**(a - 1)
     elif argindex == 1:
         a, z = self.args
         return uppergamma(a, z)*log(z) + meijerg([], [1, 1], [0, 0, a], [], z)
     else:
         raise ArgumentIndexError(self, argindex)

    def eval(cls, nu, z):
        from sympy import unpolarify, expand_mul, uppergamma, exp, gamma, factorial

        nu2 = unpolarify(nu)
        if nu != nu2:
            return expint(nu2, z)
        if nu.is_Integer and nu <= 0 or (not nu.is_Integer and (2 * nu).is_Integer):
            return unpolarify(expand_mul(z ** (nu - 1) * uppergamma(1 - nu, z)))

        # Extract branching information. This can be deduced from what is
        # explained in lowergamma.eval().
        z, n = z.extract_branch_factor()
        if n == 0:
            return
        if nu.is_integer:
            if (nu > 0) is not True:
                return
            return expint(nu, z) - 2 * pi * I * n * (-1) ** (nu - 1) / factorial(nu - 1) * unpolarify(z) ** (nu - 1)
        else:
            return (exp(2 * I * pi * nu * n) - 1) * z ** (nu - 1) * gamma(1 - nu) + expint(nu, z)

    def eval(cls, a, z):
        from sympy import unpolarify, I, expint
        if z.is_Number:
            if z is S.NaN:
                return S.NaN
            elif z is S.Infinity:
                return S.Zero
            elif z is S.Zero:
                # TODO: Holds only for Re(a) > 0:
                return gamma(a)

        # We extract branching information here. C/f lowergamma.
        nx, n = z.extract_branch_factor()
        if a.is_integer and (a > 0) == True:
            nx = unpolarify(z)
            if z != nx:
                return uppergamma(a, nx)
        elif a.is_integer and (a <= 0) == True:
            if n != 0:
                return -2*pi*I*n*(-1)**(-a)/factorial(-a) + uppergamma(a, nx)
        elif n != 0:
            return gamma(a)*(1 - exp(2*pi*I*n*a)) + exp(2*pi*I*n*a)*uppergamma(a, nx)

        # Special values.
        if a.is_Number:
            if a is S.One:
                return exp(-z)
            elif a is S.Half:
                return sqrt(pi)*erfc(sqrt(z))
            elif a.is_Integer or (2*a).is_Integer:
                b = a - 1
                if b.is_positive:
                    if a.is_integer:
                        return exp(-z) * factorial(b) * Add(*[z**k / factorial(k) for k in range(a)])
                    else:
                        return gamma(a) * erfc(sqrt(z)) + (-1)**(a - S(3)/2) * exp(-z) * sqrt(z) * Add(*[gamma(-S.Half - k) * (-z)**k / gamma(1-a) for k in range(a - S.Half)])
                elif b.is_Integer:
                    return expint(-b, z)*unpolarify(z)**(b + 1)

                if not a.is_Integer:
                    return (-1)**(S.Half - a) * pi*erfc(sqrt(z))/gamma(1-a) - z**a * exp(-z) * Add(*[z**k * gamma(a) / gamma(a+k+1) for k in range(S.Half - a)])