Python sympy.polygamma函数代码示例

def test_factorial_simplify_fail():
    # simplify(factorial(x + 1).diff(x) - ((x + 1)*factorial(x)).diff(x))) == 0
    from sympy.abc import x

    assert (
        simplify(x * polygamma(0, x + 1) - x * polygamma(0, x + 2) + polygamma(0, x + 1) - polygamma(0, x + 2) + 1) == 0
    )

def test_polygamma_expansion():
    # A. & S., pa. 259 and 260
    assert polygamma(0, 1 / x).nseries(x, n=3) == -log(x) - x / 2 - x ** 2 / 12 + O(x ** 4)
    assert polygamma(1, 1 / x).series(x, n=5) == x + x ** 2 / 2 + x ** 3 / 6 + O(x ** 5)
    assert polygamma(3, 1 / x).nseries(x, n=11) == 2 * x ** 3 + 3 * x ** 4 + 2 * x ** 5 - x ** 7 + 4 * x ** 9 / 3 + O(
        x ** 11
    )

def test_factorial_diff():
    n = Symbol('n', integer=True)

    assert factorial(n).diff(n) == \
        gamma(1 + n)*polygamma(0, 1 + n)
    assert factorial(n**2).diff(n) == \
        2*n*gamma(1 + n**2)*polygamma(0, 1 + n**2)

def test_loggamma():
    raises(TypeError, lambda: loggamma(2, 3))
    raises(ArgumentIndexError, lambda: loggamma(x).fdiff(2))
    assert loggamma(x).diff(x) == polygamma(0, x)
    s1 = loggamma(1/(x + sin(x)) + cos(x)).nseries(x, n=4)
    s2 = (-log(2*x) - 1)/(2*x) - log(x/pi)/2 + (4 - log(2*x))*x/24 + O(x**2) + \
        log(x)*x**2/2
    assert (s1 - s2).expand(force=True).removeO() == 0
    s1 = loggamma(1/x).series(x)
    s2 = (1/x - S(1)/2)*log(1/x) - 1/x + log(2*pi)/2 + \
        x/12 - x**3/360 + x**5/1260 + O(x**7)
    assert ((s1 - s2).expand(force=True)).removeO() == 0

    assert loggamma(x).rewrite('intractable') == log(gamma(x))

    s1 = loggamma(x).series(x)
    assert s1 == -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + \
        pi**4*x**4/360 + x**5*polygamma(4, 1)/120 + O(x**6)
    assert s1 == loggamma(x).rewrite('intractable').series(x)

    assert loggamma(x).is_real is None
    y, z = Symbol('y', real=True), Symbol('z', imaginary=True)
    assert loggamma(y).is_real
    assert loggamma(z).is_real is False

    def tN(N, M):
        assert loggamma(1/x)._eval_nseries(x, n=N, logx=None).getn() == M
    tN(0, 0)
    tN(1, 1)
    tN(2, 3)
    tN(3, 3)
    tN(4, 5)
    tN(5, 5)

def test_function__eval_nseries():
    n = Symbol("n")

    assert sin(x)._eval_nseries(x, 2, None) == x + O(x ** 2)
    assert sin(x + 1)._eval_nseries(x, 2, None) == x * cos(1) + sin(1) + O(x ** 2)
    assert sin(pi * (1 - x))._eval_nseries(x, 2, None) == pi * x + O(x ** 2)
    assert acos(1 - x ** 2)._eval_nseries(x, 2, None) == sqrt(2) * x + O(x ** 2)
    assert polygamma(n, x + 1)._eval_nseries(x, 2, None) == polygamma(n, 1) + polygamma(n + 1, 1) * x + O(x ** 2)
    raises(PoleError, "sin(1/x)._eval_nseries(x,2,None)")
    raises(PoleError, "acos(1-x)._eval_nseries(x,2,None)")
    raises(PoleError, "acos(1+x)._eval_nseries(x,2,None)")
    assert loggamma(1 / x)._eval_nseries(x, 0, None) == log(x) / 2 - log(x) / x - 1 / x + O(1, x)
    assert loggamma(log(1 / x)).nseries(x, n=1, logx=y) == loggamma(-y)

    def fdiff(self, argindex=1):
        from sympy import polygamma

        if argindex == 1:
            # http://functions.wolfram.com/GammaBetaErf/Binomial/20/01/01/
            n, k = self.args
            return binomial(n, k) * (polygamma(0, n + 1) - polygamma(0, n - k + 1))
        elif argindex == 2:
            # http://functions.wolfram.com/GammaBetaErf/Binomial/20/01/02/
            n, k = self.args
            return binomial(n, k) * (polygamma(0, n - k + 1) - polygamma(0, k + 1))
        else:
            raise ArgumentIndexError(self, argindex)

def test_catalan():
    assert catalan(1) == 1
    assert catalan(2) == 2
    assert catalan(3) == 5
    assert catalan(4) == 14

    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(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 diff(catalan(x),x) == (polygamma(0, x + Rational(1,2)) - polygamma(0, x + 2) + 2*log(2))*catalan(x)

    c = catalan(0.5).evalf()
    assert str(c) == '0.848826363156775'

def test_FunctionWrapper():
    import sympy
    n, m, theta, phi = sympy.symbols("n, m, theta, phi")
    r = sympy.Ynm(n, m, theta, phi)
    s = Integer(2)*r
    assert isinstance(s, Mul)
    assert isinstance(s.args[1]._sympy_(), sympy.Ynm)

    x = symbols("x")
    e = x + sympy.loggamma(x)
    assert str(e) == "x + loggamma(x)"
    assert isinstance(e, Add)
    assert e + sympy.loggamma(x) == x + 2*sympy.loggamma(x)

    f = e.subs({x : 10})
    assert f == 10 + log(362880)

    f = e.subs({x : 2})
    assert f == 2

    f = e.subs({x : 100});
    v = f.n(53, real=True);
    assert abs(float(v) - 459.13420537) < 1e-7

    f = e.diff(x)
    assert f == 1 + sympy.polygamma(0, x)

def test_gamma_series():
    assert gamma(x + 1).series(x, 0, 3) == \
        1 - EulerGamma*x + x**2*(EulerGamma**2/2 + pi**2/12) + O(x**3)
    assert gamma(x).series(x, -1, 3) == \
        -1/(x + 1) + EulerGamma - 1 + (x + 1)*(-1 - pi**2/12 - EulerGamma**2/2 + \
       EulerGamma) + (x + 1)**2*(-1 - pi**2/12 - EulerGamma**2/2 + EulerGamma**3/6 - \
       polygamma(2, 1)/6 + EulerGamma*pi**2/12 + EulerGamma) + O((x + 1)**3, (x, -1))

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

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_gamma_series():
    assert gamma(x + 1).series(x, 0, 3) == \
        1 - EulerGamma*x + x**2*(EulerGamma**2/2 + pi**2/12) + O(x**3)
    assert gamma(x).series(x, -1, 3) == \
        -1/x + EulerGamma - 1 + x*(-1 - pi**2/12 - EulerGamma**2/2 + EulerGamma) \
        + x**2*(-1 - pi**2/12 - EulerGamma**2/2 + EulerGamma**3/6 -
        polygamma(2, 1)/6 + EulerGamma*pi**2/12 + EulerGamma) + O(x**3)

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 test_evalf_default():
    from sympy.functions.special.gamma_functions import polygamma
    assert type(sin(4.0)) == Real
    assert type(re(sin(I + 1.0))) == Real
    assert type(im(sin(I + 1.0))) == Real
    assert type(sin(4)) == sin
    assert type(polygamma(2,4.0)) == Real
    assert type(sin(Rational(1,4))) == sin

def test_gamma_series():
    assert gamma(x + 1).series(x, 0, 3) == \
        1 - x*EulerGamma + x**2*EulerGamma**2/2 + pi**2*x**2/12 + O(x**3)
    assert gamma(x).series(x, -1, 3) == \
        -1/x + EulerGamma - 1 + EulerGamma*x - EulerGamma**2*x/2 - pi**2*x/12 \
        - x + EulerGamma*x**2 + EulerGamma*pi**2*x**2/12 - \
        x**2*polygamma(2, 1)/6 + EulerGamma**3*x**2/6 - EulerGamma**2*x**2/2 \
        - pi**2*x**2/12 - x**2 + O(x**3)