Python sympy.exp_polar函数代码示例

    def eval(cls, arg):
        from sympy import exp_polar, pi, I, arg as argument
        if arg.is_number:
            ar = argument(arg)
            #if not ar.has(argument) and not ar.has(atan):
            if ar in (0, pi/2, -pi/2, pi):
                return exp_polar(I*ar)*abs(arg)

        if arg.is_Mul:
            args = arg.args
        else:
            args = [arg]
        included = []
        excluded = []
        positive = []
        for arg in args:
            if arg.is_polar:
                included += [arg]
            elif arg.is_positive:
                positive += [arg]
            else:
                excluded += [arg]
        if len(excluded) < len(args):
            if excluded:
                return Mul(*(included + positive))*polar_lift(Mul(*excluded))
            elif included:
                return Mul(*(included + positive))
            else:
                return Mul(*positive)*exp_polar(0)

    def t(func, hyp, z):
        """ Test that func is a valid representation of hyp. """
        # First test that func agrees with hyp for small z
        if not tn(func.rewrite('nonrepsmall'), hyp, z,
                  a=S(-1)/2, b=S(-1)/2, c=S(1)/2, d=S(1)/2):
            return False
        # Next check that the two small representations agree.
        if not tn(
            func.rewrite('nonrepsmall').subs(
                z, exp_polar(I*pi)*z).replace(exp_polar, exp),
            func.subs(z, exp_polar(I*pi)*z).rewrite('nonrepsmall'),
                z, a=S(-1)/2, b=S(-1)/2, c=S(1)/2, d=S(1)/2):
            return False
        # Next check continuity along exp_polar(I*pi)*t
        expr = func.subs(z, exp_polar(I*pi)*z).rewrite('nonrep')
        if abs(expr.subs(z, 1 + 1e-15).n() - expr.subs(z, 1 - 1e-15).n()) > 1e-10:
            return False
        # Finally check continuity of the big reps.

        def dosubs(func, a, b):
            rv = func.subs(z, exp_polar(a)*z).rewrite('nonrep')
            return rv.subs(z, exp_polar(b)*z).replace(exp_polar, exp)
        for n in [0, 1, 2, 3, 4, -1, -2, -3, -4]:
            expr1 = dosubs(func, 2*I*pi*n, I*pi/2)
            expr2 = dosubs(func, 2*I*pi*n + I*pi, -I*pi/2)
            if not tn(expr1, expr2, z):
                return False
            expr1 = dosubs(func, 2*I*pi*(n + 1), -I*pi/2)
            expr2 = dosubs(func, 2*I*pi*n + I*pi, I*pi/2)
            if not tn(expr1, expr2, z):
                return False
        return True

def test_ci():
    m1 = exp_polar(I * pi)
    m1_ = exp_polar(-I * pi)
    pI = exp_polar(I * pi / 2)
    mI = exp_polar(-I * pi / 2)

    assert Ci(m1 * x) == Ci(x) + I * pi
    assert Ci(m1_ * x) == Ci(x) - I * pi
    assert Ci(pI * x) == Chi(x) + I * pi / 2
    assert Ci(mI * x) == Chi(x) - I * pi / 2
    assert Chi(m1 * x) == Chi(x) + I * pi
    assert Chi(m1_ * x) == Chi(x) - I * pi
    assert Chi(pI * x) == Ci(x) + I * pi / 2
    assert Chi(mI * x) == Ci(x) - I * pi / 2
    assert Ci(exp_polar(2 * I * pi) * x) == Ci(x) + 2 * I * pi
    assert Chi(exp_polar(-2 * I * pi) * x) == Chi(x) - 2 * I * pi
    assert Chi(exp_polar(2 * I * pi) * x) == Chi(x) + 2 * I * pi
    assert Ci(exp_polar(-2 * I * pi) * x) == Ci(x) - 2 * I * pi

    assert mytd(Ci(x), cos(x) / x, x)
    assert mytd(Chi(x), cosh(x) / x, x)

    assert mytn(Ci(x), Ci(x).rewrite(Ei), Ei(x * exp_polar(-I * pi / 2)) / 2 + Ei(x * exp_polar(I * pi / 2)) / 2, x)
    assert mytn(Chi(x), Chi(x).rewrite(Ei), Ei(x) / 2 + Ei(x * exp_polar(I * pi)) / 2 - I * pi / 2, x)

    assert tn_arg(Ci)
    assert tn_arg(Chi)

    from sympy import O, EulerGamma, log, limit

    assert Ci(x).nseries(x, n=4) == EulerGamma + log(x) - x ** 2 / 4 + x ** 4 / 96 + O(x ** 5)
    assert Chi(x).nseries(x, n=4) == EulerGamma + log(x) + x ** 2 / 4 + x ** 4 / 96 + O(x ** 5)
    assert limit(log(x) - Ci(2 * x), x, 0) == -log(2) - EulerGamma

def test_si():
    assert Si(I*x) == I*Shi(x)
    assert Shi(I*x) == I*Si(x)
    assert Si(-I*x) == -I*Shi(x)
    assert Shi(-I*x) == -I*Si(x)
    assert Si(-x) == -Si(x)
    assert Shi(-x) == -Shi(x)
    assert Si(exp_polar(2*pi*I)*x) == Si(x)
    assert Si(exp_polar(-2*pi*I)*x) == Si(x)
    assert Shi(exp_polar(2*pi*I)*x) == Shi(x)
    assert Shi(exp_polar(-2*pi*I)*x) == Shi(x)

    assert mytd(Si(x), sin(x)/x, x)
    assert mytd(Shi(x), sinh(x)/x, x)

    assert mytn(Si(x), Si(x).rewrite(Ei),
                -I*(-Ei(x*exp_polar(-I*pi/2))/2 \
                        + Ei(x*exp_polar(I*pi/2))/2 - I*pi) + pi/2, x)
    assert mytn(Si(x), Si(x).rewrite(expint),
                -I*(-expint(1, x*exp_polar(-I*pi/2))/2 + \
                    expint(1, x*exp_polar(I*pi/2))/2) + pi/2, x)
    assert mytn(Shi(x), Shi(x).rewrite(Ei),
                Ei(x)/2 - Ei(x*exp_polar(I*pi))/2 + I*pi/2, x)
    assert mytn(Shi(x), Shi(x).rewrite(expint),
                expint(1, x)/2 - expint(1, x*exp_polar(I*pi))/2 - I*pi/2, x)

    assert tn_arg(Si)
    assert tn_arg(Shi)

    from sympy import O
    assert Si(x).nseries(x, n=8) == x - x**3/18 + x**5/600 - x**7/35280 + O(x**9)
    assert Shi(x).nseries(x, n=8) == x + x**3/18 + x**5/600 + x**7/35280 + O(x**9)
    assert Si(sin(x)).nseries(x, n=5) == x - 2*x**3/9 + 17*x**5/450 + O(x**6)
    assert Si(x).nseries(x, 1, n=3) == \
           Si(1) + x*sin(1) + x**2/2*cos(1) - x**2/2*sin(1) + O(x**3)

def test_uppergamma():
    from sympy import meijerg, exp_polar, I, expint
    assert uppergamma(4, 0) == 6
    assert uppergamma(x, y).diff(y) == -y**(x-1)*exp(-y)
    assert td(uppergamma(randcplx(), y), y)
    assert uppergamma(x, y).diff(x) == \
           uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
    assert td(uppergamma(x, randcplx()), x)

    assert uppergamma(S.Half, x) == sqrt(pi)*(1 - erf(sqrt(x)))
    assert not uppergamma(S.Half - 3, x).has(uppergamma)
    assert not uppergamma(S.Half + 3, x).has(uppergamma)
    assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
    assert tn(uppergamma(S.Half + 3, x, evaluate=False),
              uppergamma(S.Half + 3, x), x)
    assert tn(uppergamma(S.Half - 3, x, evaluate=False),
              uppergamma(S.Half - 3, x), x)

    assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)

    assert tn_branch(-3, uppergamma)
    assert tn_branch(-4, uppergamma)
    assert tn_branch(S(1)/3, uppergamma)
    assert tn_branch(pi, uppergamma)
    assert uppergamma(3, exp_polar(4*pi*I)*x) == uppergamma(3, x)
    assert uppergamma(y, exp_polar(5*pi*I)*x) == \
           exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + gamma(y)*(1-exp(4*pi*I*y))
    assert uppergamma(-2, exp_polar(5*pi*I)*x) == \
           uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I

    assert uppergamma(-2, x) == expint(3, x)/x**2
    assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y)

def test_lowergamma():
    from sympy import meijerg, exp_polar, I, expint
    assert lowergamma(x, y).diff(y) == y**(x-1)*exp(-y)
    assert td(lowergamma(randcplx(), y), y)
    assert lowergamma(x, y).diff(x) == \
           gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
           + meijerg([], [1, 1], [0, 0, x], [], y)

    assert lowergamma(S.Half, x) == sqrt(pi)*erf(sqrt(x))
    assert not lowergamma(S.Half - 3, x).has(lowergamma)
    assert not lowergamma(S.Half + 3, x).has(lowergamma)
    assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
    assert tn(lowergamma(S.Half + 3, x, evaluate=False),
              lowergamma(S.Half + 3, x), x)
    assert tn(lowergamma(S.Half - 3, x, evaluate=False),
              lowergamma(S.Half - 3, x), x)

    assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)

    assert tn_branch(-3, lowergamma)
    assert tn_branch(-4, lowergamma)
    assert tn_branch(S(1)/3, lowergamma)
    assert tn_branch(pi, lowergamma)
    assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x)
    assert lowergamma(y, exp_polar(5*pi*I)*x) == \
           exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
    assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
           lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I

    assert lowergamma(x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x)
    k = Symbol('k', integer=True)
    assert lowergamma(k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k)
    k = Symbol('k', integer=True, positive=False)
    assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)

def test_ei():
    pos = Symbol('p', positive=True)
    neg = Symbol('n', negative=True)
    assert Ei(-pos) == Ei(polar_lift(-1)*pos) - I*pi
    assert Ei(neg) == Ei(polar_lift(neg)) - I*pi
    assert tn_branch(Ei)
    assert mytd(Ei(x), exp(x)/x, x)
    assert mytn(Ei(x), Ei(x).rewrite(uppergamma),
                -uppergamma(0, x*polar_lift(-1)) - I*pi, x)
    assert mytn(Ei(x), Ei(x).rewrite(expint),
                -expint(1, x*polar_lift(-1)) - I*pi, x)
    assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x)
    assert Ei(x*exp_polar(2*I*pi)) == Ei(x) + 2*I*pi
    assert Ei(x*exp_polar(-2*I*pi)) == Ei(x) - 2*I*pi

    assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x)
    assert mytn(Ei(x*polar_lift(I)), Ei(x*polar_lift(I)).rewrite(Si),
                Ci(x) + I*Si(x) + I*pi/2, x)

    assert Ei(log(x)).rewrite(li) == li(x)
    assert Ei(2*log(x)).rewrite(li) == li(x**2)

    assert gruntz(Ei(x+exp(-x))*exp(-x)*x, x, oo) == 1

    assert Ei(x).series(x) == EulerGamma + log(x) + x + x**2/4 + \
        x**3/18 + x**4/96 + x**5/600 + O(x**6)

def test_branching():
    from sympy import exp_polar, polar_lift, Symbol, I, exp
    assert besselj(polar_lift(k), x) == besselj(k, x)
    assert besseli(polar_lift(k), x) == besseli(k, x)

    n = Symbol('n', integer=True)
    assert besselj(n, exp_polar(2*pi*I)*x) == besselj(n, x)
    assert besselj(n, polar_lift(x)) == besselj(n, x)
    assert besseli(n, exp_polar(2*pi*I)*x) == besseli(n, x)
    assert besseli(n, polar_lift(x)) == besseli(n, x)

    def tn(func, s):
        from random import uniform
        c = uniform(1, 5)
        expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
        eps = 1e-15
        expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
        return abs(expr.n() - expr2.n()).n() < 1e-10

    nu = Symbol('nu')
    assert besselj(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besselj(nu, x)
    assert besseli(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besseli(nu, x)
    assert tn(besselj, 2)
    assert tn(besselj, pi)
    assert tn(besselj, I)
    assert tn(besseli, 2)
    assert tn(besseli, pi)
    assert tn(besseli, I)

 def tn(func, s):
     from random import uniform
     c = uniform(1, 5)
     expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
     eps = 1e-15
     expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
     return abs(expr.n() - expr2.n()).n() < 1e-10

def test_issue_7173():
    assert laplace_transform(sinh(a*x)*cosh(a*x), x, s) == \
        (a/(s**2 - 4*a**2), 0,
        And(Or(Abs(periodic_argument(exp_polar(I*pi)*polar_lift(a), oo)) <
        pi/2, Abs(periodic_argument(exp_polar(I*pi)*polar_lift(a), oo)) <=
        pi/2), Or(Abs(periodic_argument(a, oo)) < pi/2,
        Abs(periodic_argument(a, oo)) <= pi/2)))

def test_meijerg_eval():
    from sympy import besseli, exp_polar
    from sympy.abc import l
    a = randcplx()
    arg = x*exp_polar(k*pi*I)
    expr1 = pi*meijerg([[], [(a + 1)/2]], [[a/2], [-a/2, (a + 1)/2]], arg**2/4)
    expr2 = besseli(a, arg)

    # Test that the two expressions agree for all arguments.
    for x_ in [0.5, 1.5]:
        for k_ in [0.0, 0.1, 0.3, 0.5, 0.8, 1, 5.751, 15.3]:
            assert abs((expr1 - expr2).n(subs={x: x_, k: k_})) < 1e-10
            assert abs((expr1 - expr2).n(subs={x: x_, k: -k_})) < 1e-10

    # Test continuity independently
    eps = 1e-13
    expr2 = expr1.subs(k, l)
    for x_ in [0.5, 1.5]:
        for k_ in [0.5, S(1)/3, 0.25, 0.75, S(2)/3, 1.0, 1.5]:
            assert abs((expr1 - expr2).n(
                       subs={x: x_, k: k_ + eps, l: k_ - eps})) < 1e-10
            assert abs((expr1 - expr2).n(
                       subs={x: x_, k: -k_ + eps, l: -k_ - eps})) < 1e-10

    expr = (meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(-I*pi)/4)
            + meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(I*pi)/4)) \
        /(2*sqrt(pi))
    assert (expr - pi/exp(1)).n(chop=True) == 0

def test_expint():
    assert mytn(expint(x, y), expint(x, y).rewrite(uppergamma), y ** (x - 1) * uppergamma(1 - x, y), x)
    assert mytd(expint(x, y), -y ** (x - 1) * meijerg([], [1, 1], [0, 0, 1 - x], [], y), x)
    assert mytd(expint(x, y), -expint(x - 1, y), y)
    assert mytn(expint(1, x), expint(1, x).rewrite(Ei), -Ei(x * polar_lift(-1)) + I * pi, x)

    assert (
        expint(-4, x)
        == exp(-x) / x + 4 * exp(-x) / x ** 2 + 12 * exp(-x) / x ** 3 + 24 * exp(-x) / x ** 4 + 24 * exp(-x) / x ** 5
    )
    assert expint(-S(3) / 2, x) == exp(-x) / x + 3 * exp(-x) / (2 * x ** 2) - 3 * sqrt(pi) * erf(sqrt(x)) / (
        4 * x ** S("5/2")
    ) + 3 * sqrt(pi) / (4 * x ** S("5/2"))

    assert tn_branch(expint, 1)
    assert tn_branch(expint, 2)
    assert tn_branch(expint, 3)
    assert tn_branch(expint, 1.7)
    assert tn_branch(expint, pi)

    assert expint(y, x * exp_polar(2 * I * pi)) == x ** (y - 1) * (exp(2 * I * pi * y) - 1) * gamma(-y + 1) + expint(
        y, x
    )
    assert expint(y, x * exp_polar(-2 * I * pi)) == x ** (y - 1) * (exp(-2 * I * pi * y) - 1) * gamma(-y + 1) + expint(
        y, x
    )
    assert expint(2, x * exp_polar(2 * I * pi)) == 2 * I * pi * x + expint(2, x)
    assert expint(2, x * exp_polar(-2 * I * pi)) == -2 * I * pi * x + expint(2, x)
    assert expint(1, x).rewrite(Ei).rewrite(expint) == expint(1, x)

    assert mytn(E1(x), E1(x).rewrite(Shi), Shi(x) - Chi(x), x)
    assert mytn(E1(polar_lift(I) * x), E1(polar_lift(I) * x).rewrite(Si), -Ci(x) + I * Si(x) - I * pi / 2, x)

    assert mytn(expint(2, x), expint(2, x).rewrite(Ei).rewrite(expint), -x * E1(x) + exp(-x), x)
    assert mytn(expint(3, x), expint(3, x).rewrite(Ei).rewrite(expint), x ** 2 * E1(x) / 2 + (1 - x) * exp(-x) / 2, x)

def test_gamma():
    assert gamma(nan) == nan
    assert gamma(oo) == oo

    assert gamma(-100) == zoo
    assert gamma(0) == zoo

    assert gamma(1) == 1
    assert gamma(2) == 1
    assert gamma(3) == 2

    assert gamma(102) == factorial(101)

    assert gamma(Rational(1, 2)) == sqrt(pi)

    assert gamma(Rational(3, 2)) == Rational(1, 2)*sqrt(pi)
    assert gamma(Rational(5, 2)) == Rational(3, 4)*sqrt(pi)
    assert gamma(Rational(7, 2)) == Rational(15, 8)*sqrt(pi)

    assert gamma(Rational(-1, 2)) == -2*sqrt(pi)
    assert gamma(Rational(-3, 2)) == Rational(4, 3)*sqrt(pi)
    assert gamma(Rational(-5, 2)) == -Rational(8, 15)*sqrt(pi)

    assert gamma(Rational(-15, 2)) == Rational(256, 2027025)*sqrt(pi)

    assert gamma(Rational(
        -11, 8)).expand(func=True) == Rational(64, 33)*gamma(Rational(5, 8))
    assert gamma(Rational(
        -10, 3)).expand(func=True) == Rational(81, 280)*gamma(Rational(2, 3))
    assert gamma(Rational(
        14, 3)).expand(func=True) == Rational(880, 81)*gamma(Rational(2, 3))
    assert gamma(Rational(
        17, 7)).expand(func=True) == Rational(30, 49)*gamma(Rational(3, 7))
    assert gamma(Rational(
        19, 8)).expand(func=True) == Rational(33, 64)*gamma(Rational(3, 8))

    assert gamma(x).diff(x) == gamma(x)*polygamma(0, x)

    assert gamma(x - 1).expand(func=True) == gamma(x)/(x - 1)
    assert gamma(x + 2).expand(func=True, mul=False) == x*(x + 1)*gamma(x)

    assert conjugate(gamma(x)) == gamma(conjugate(x))

    assert expand_func(gamma(x + Rational(3, 2))) == \
        (x + Rational(1, 2))*gamma(x + Rational(1, 2))

    assert expand_func(gamma(x - Rational(1, 2))) == \
        gamma(Rational(1, 2) + x)/(x - Rational(1, 2))

    # Test a bug:
    assert expand_func(gamma(x + Rational(3, 4))) == gamma(x + Rational(3, 4))

    assert gamma(3*exp_polar(I*pi)/4).is_nonnegative is False
    assert gamma(3*exp_polar(I*pi)/4).is_nonpositive is True

    # Issue 8526
    k = Symbol('k', integer=True, nonnegative=True)
    assert isinstance(gamma(k), gamma)
    assert gamma(-k) == zoo

def tn_branch(s, func):
    from sympy import I, pi, exp_polar
    from random import uniform
    c = uniform(1, 5)
    expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
    eps = 1e-15
    expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
    return abs(expr.n() - expr2.n()).n() < 1e-10

def test_hyper_unpolarify():
    from sympy import exp_polar
    a = exp_polar(2*pi*I)*x
    b = x
    assert hyper([], [], a).argument == b
    assert hyper([0], [], a).argument == a
    assert hyper([0], [0], a).argument == b
    assert hyper([0, 1], [0], a).argument == a
    assert hyper([0, 1], [0], exp_polar(2*pi*I)).argument == 1