Ejemplo n.º 1
0
def test_manualintegrate_special():
    f, F = 4*exp(-x**2/3), 2*sqrt(3)*sqrt(pi)*erf(sqrt(3)*x/3)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = 3*exp(4*x**2), 3*sqrt(pi)*erfi(2*x)/4
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = x**Rational(1, 3)*exp(-x/8), -16*uppergamma(Rational(4, 3), x/8)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = exp(2*x)/x, Ei(2*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = exp(1 + 2*x - x**2), sqrt(pi)*exp(2)*erf(x - 1)/2
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f = sin(x**2 + 4*x + 1)
    F = (sqrt(2)*sqrt(pi)*(-sin(3)*fresnelc(sqrt(2)*(2*x + 4)/(2*sqrt(pi))) +
        cos(3)*fresnels(sqrt(2)*(2*x + 4)/(2*sqrt(pi))))/2)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = cos(4*x**2), sqrt(2)*sqrt(pi)*fresnelc(2*sqrt(2)*x/sqrt(pi))/4
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = sin(3*x + 2)/x, sin(2)*Ci(3*x) + cos(2)*Si(3*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = sinh(3*x - 2)/x, -sinh(2)*Chi(3*x) + cosh(2)*Shi(3*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = 5*cos(2*x - 3)/x, 5*cos(3)*Ci(2*x) + 5*sin(3)*Si(2*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = cosh(x/2)/x, Chi(x/2)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = cos(x**2)/x, Ci(x**2)/2
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = 1/log(2*x + 1), li(2*x + 1)/2
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = polylog(2, 5*x)/x, polylog(3, 5*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = 5/sqrt(3 - 2*sin(x)**2), 5*sqrt(3)*elliptic_f(x, Rational(2, 3))/3
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = sqrt(4 + 9*sin(x)**2), 2*elliptic_e(x, Rational(-9, 4))
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
Ejemplo n.º 2
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def test_lerchphi_expansion():
    assert myexpand(lerchphi(1, s, a), zeta(s, a))
    assert myexpand(lerchphi(z, s, 1), polylog(s, z) / z)

    # direct summation
    assert myexpand(lerchphi(z, -1, a), a / (1 - z) + z / (1 - z)**2)
    assert myexpand(lerchphi(z, -3, a), None)
    # polylog reduction
    assert myexpand(
        lerchphi(z, s, S.Half),
        2**(s - 1) * (polylog(s, sqrt(z)) / sqrt(z) -
                      polylog(s,
                              polar_lift(-1) * sqrt(z)) / sqrt(z)))
    assert myexpand(lerchphi(z, s, 2), -1 / z + polylog(s, z) / z**2)
    assert myexpand(lerchphi(z, s, Rational(3, 2)), None)
    assert myexpand(lerchphi(z, s, Rational(7, 3)), None)
    assert myexpand(lerchphi(z, s, Rational(-1, 3)), None)
    assert myexpand(lerchphi(z, s, Rational(-5, 2)), None)

    # hurwitz zeta reduction
    assert myexpand(lerchphi(-1, s, a),
                    2**(-s) * zeta(s, a / 2) - 2**(-s) * zeta(s, (a + 1) / 2))
    assert myexpand(lerchphi(I, s, a), None)
    assert myexpand(lerchphi(-I, s, a), None)
    assert myexpand(lerchphi(exp(I * pi * Rational(2, 5)), s, a), None)
Ejemplo n.º 3
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def test_lerchphi_expansion():
    assert myexpand(lerchphi(1, s, a), zeta(s, a))
    assert myexpand(lerchphi(z, s, 1), polylog(s, z) / z)

    # direct summation
    assert myexpand(lerchphi(z, -1, a), a / (1 - z) + z / (1 - z)**2)
    assert myexpand(lerchphi(z, -3, a), None)

    # polylog reduction
    assert myexpand(
        lerchphi(z, s,
                 S(1) / 2),
        2**(s - 1) * (polylog(s, sqrt(z)) / sqrt(z) -
                      polylog(s,
                              polar_lift(-1) * sqrt(z)) / sqrt(z)))
    assert myexpand(lerchphi(z, s, 2), -1 / z + polylog(s, z) / z**2)
    assert myexpand(lerchphi(z, s, S(3) / 2), None)
    assert myexpand(lerchphi(z, s, S(7) / 3), None)
    assert myexpand(lerchphi(z, s, -S(1) / 3), None)
    assert myexpand(lerchphi(z, s, -S(5) / 2), None)

    # hurwitz zeta reduction
    assert myexpand(lerchphi(-1, s, a),
                    2**(-s) * zeta(s, a / 2) - 2**(-s) * zeta(s, (a + 1) / 2))
    assert myexpand(lerchphi(I, s, a), None)
    assert myexpand(lerchphi(-I, s, a), None)
    assert myexpand(lerchphi(exp(2 * I * pi / 5), s, a), None)
Ejemplo n.º 4
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def test_manualintegrate_special():
    f, F = 4*exp(-x**2/3), 2*sqrt(3)*sqrt(pi)*erf(sqrt(3)*x/3)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = 3*exp(4*x**2), 3*sqrt(pi)*erfi(2*x)/4
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = x**(S(1)/3)*exp(-x/8), -16*uppergamma(S(4)/3, x/8)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = exp(2*x)/x, Ei(2*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = exp(1 + 2*x - x**2), sqrt(pi)*exp(2)*erf(x - 1)/2
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f = sin(x**2 + 4*x + 1)
    F = (sqrt(2)*sqrt(pi)*(-sin(3)*fresnelc(sqrt(2)*(2*x + 4)/(2*sqrt(pi))) +
        cos(3)*fresnels(sqrt(2)*(2*x + 4)/(2*sqrt(pi))))/2)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = cos(4*x**2), sqrt(2)*sqrt(pi)*fresnelc(2*sqrt(2)*x/sqrt(pi))/4
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = sin(3*x + 2)/x, sin(2)*Ci(3*x) + cos(2)*Si(3*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = sinh(3*x - 2)/x, -sinh(2)*Chi(3*x) + cosh(2)*Shi(3*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = 5*cos(2*x - 3)/x, 5*cos(3)*Ci(2*x) + 5*sin(3)*Si(2*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = cosh(x/2)/x, Chi(x/2)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = cos(x**2)/x, Ci(x**2)/2
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = 1/log(2*x + 1), li(2*x + 1)/2
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = polylog(2, 5*x)/x, polylog(3, 5*x)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = 5/sqrt(3 - 2*sin(x)**2), 5*sqrt(3)*elliptic_f(x, S(2)/3)/3
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
    f, F = sqrt(4 + 9*sin(x)**2), 2*elliptic_e(x, -S(9)/4)
    assert manualintegrate(f, x) == F and F.diff(x).equals(f)
Ejemplo n.º 5
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def test_issue_10798():
    from sympy import integrate, pi, I, log, polylog, exp_polar, Piecewise, meijerg, Abs
    from sympy.abc import x, y
    assert integrate(1/(1-(x*y)**2), (x, 0, 1), y) == \
        -Piecewise((I*pi*log(y) - polylog(2, y), Abs(y) < 1), (-I*pi*log(1/y) - polylog(2, y), Abs(1/y) < 1), \
                   (-I*pi*meijerg(((), (1, 1)), ((0, 0), ()), y) + I*pi*meijerg(((1, 1), ()), ((), (0, 0)), y) - polylog(2, y), True))/2 \
                   - log(y)*log(1 - 1/y)/2 + log(y)*log(1 + 1/y)/2 + log(y)*log(y - 1)/2 \
                   - log(y)*log(y + 1)/2 + I*pi*log(y)/2 - polylog(2, y*exp_polar(I*pi))/2
Ejemplo n.º 6
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def test_issue_10798():
    from sympy import integrate, pi, I, log, polylog, exp_polar, Piecewise, meijerg, Abs
    from sympy.abc import x, y
    assert integrate(1/(1-(x*y)**2), (x, 0, 1), y) == \
        -Piecewise((I*pi*log(y) - polylog(2, y), Abs(y) < 1), (-I*pi*log(1/y) - polylog(2, y), Abs(1/y) < 1), \
                   (-I*pi*meijerg(((), (1, 1)), ((0, 0), ()), y) + I*pi*meijerg(((1, 1), ()), ((), (0, 0)), y) - polylog(2, y), True))/2 \
                   - log(y)*log(1 - 1/y)/2 + log(y)*log(1 + 1/y)/2 + log(y)*log(y - 1)/2 \
                   - log(y)*log(y + 1)/2 + I*pi*log(y)/2 - polylog(2, y*exp_polar(I*pi))/2
Ejemplo n.º 7
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def test_polylog_series():
    assert polylog(1, z).series(
        z, n=5) == z + z**2 / 2 + z**3 / 3 + z**4 / 4 + O(z**5)
    assert polylog(1, sqrt(z)).series(z, n=3) == z/2 + z**2/4 + sqrt(z)\
        + z**(S(3)/2)/3 + z**(S(5)/2)/5 + O(z**3)

    # https://github.com/sympy/sympy/issues/9497
    assert polylog(S(3)/2, -z).series(z, 0, 5) == -z + sqrt(2)*z**2/4\
        - sqrt(3)*z**3/9 + z**4/8 + O(z**5)
Ejemplo n.º 8
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def test_lerchphi():
    from sympy import gammasimp, exp_polar, polylog, log, lerchphi

    assert hyperexpand(hyper([1, a], [a + 1], z) / a) == lerchphi(z, 1, a)
    assert hyperexpand(hyper([1, a, a], [a + 1, a + 1], z) / a**2) == lerchphi(
        z, 2, a)
    assert hyperexpand(hyper([1, a, a, a], [a + 1, a + 1, a + 1], z) /
                       a**3) == lerchphi(z, 3, a)
    assert hyperexpand(hyper([1] + [a] * 10, [a + 1] * 10, z) /
                       a**10) == lerchphi(z, 10, a)
    assert gammasimp(
        hyperexpand(meijerg([0, 1 - a], [], [0], [-a],
                            exp_polar(-I * pi) * z))) == lerchphi(z, 1, a)
    assert gammasimp(
        hyperexpand(
            meijerg([0, 1 - a, 1 - a], [], [0], [-a, -a],
                    exp_polar(-I * pi) * z))) == lerchphi(z, 2, a)
    assert gammasimp(
        hyperexpand(
            meijerg([0, 1 - a, 1 - a, 1 - a], [], [0], [-a, -a, -a],
                    exp_polar(-I * pi) * z))) == lerchphi(z, 3, a)

    assert hyperexpand(z * hyper([1, 1], [2], z)) == -log(1 + -z)
    assert hyperexpand(z * hyper([1, 1, 1], [2, 2], z)) == polylog(2, z)
    assert hyperexpand(z * hyper([1, 1, 1, 1], [2, 2, 2], z)) == polylog(3, z)

    assert hyperexpand(
        hyper([1, a, 1 + S.Half], [a + 1, S.Half],
              z)) == -2 * a / (z - 1) + (-2 * a**2 + a) * lerchphi(z, 1, a)

    # Now numerical tests. These make sure reductions etc are carried out
    # correctly

    # a rational function (polylog at negative integer order)
    assert can_do([2, 2, 2], [1, 1])

    # NOTE these contain log(1-x) etc ... better make sure we have |z| < 1
    # reduction of order for polylog
    assert can_do([1, 1, 1, b + 5], [2, 2, b], div=10)

    # reduction of order for lerchphi
    # XXX lerchphi in mpmath is flaky
    assert can_do([1, a, a, a, b + 5], [a + 1, a + 1, a + 1, b],
                  numerical=False)

    # test a bug
    from sympy import Abs

    assert hyperexpand(
        hyper(
            [S.Half, S.Half, S.Half, 1],
            [Rational(3, 2), Rational(3, 2),
             Rational(3, 2)],
            Rational(1, 4),
        )) == Abs(-polylog(3,
                           exp_polar(I * pi) / 2) + polylog(3, S.Half))
Ejemplo n.º 9
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def test_polylog_values():
    assert polylog(2, 2) == pi**2 / 4 - I * pi * log(2)
    assert polylog(2, S.Half) == pi**2 / 12 - log(2)**2 / 2
    for z in [
            S.Half, 2, (sqrt(5) - 1) / 2, -(sqrt(5) - 1) / 2,
            -(sqrt(5) + 1) / 2, (3 - sqrt(5)) / 2
    ]:
        assert Abs(
            polylog(2, z).evalf() -
            polylog(2, z, evaluate=False).evalf()) < 1e-15
    z = Symbol("z")
    for s in [-1, 0]:
        for _ in range(10):
            assert verify_numerically(polylog(s, z),
                                      polylog(s, z, evaluate=False),
                                      z,
                                      a=-3,
                                      b=-2,
                                      c=S.Half,
                                      d=2)
            assert verify_numerically(polylog(s, z),
                                      polylog(s, z, evaluate=False),
                                      z,
                                      a=2,
                                      b=-2,
                                      c=5,
                                      d=2)

    from sympy import Integral
    assert polylog(0, Integral(1, (x, 0, 1))) == -S.Half
Ejemplo n.º 10
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def test_rewriting():
    assert dirichlet_eta(x).rewrite(zeta) == (1 - 2**(1 - x))*zeta(x)
    assert zeta(x).rewrite(dirichlet_eta) == dirichlet_eta(x)/(1 - 2**(1 - x))
    assert tn(dirichlet_eta(x), dirichlet_eta(x).rewrite(zeta), x)
    assert tn(zeta(x), zeta(x).rewrite(dirichlet_eta), x)

    assert zeta(x, a).rewrite(lerchphi) == lerchphi(1, x, a)
    assert polylog(s, z).rewrite(lerchphi) == lerchphi(z, s, 1)*z

    assert lerchphi(1, x, a).rewrite(zeta) == zeta(x, a)
    assert z*lerchphi(z, s, 1).rewrite(polylog) == polylog(s, z)
Ejemplo n.º 11
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def test_rewriting():
    assert dirichlet_eta(x).rewrite(zeta) == (1 - 2 ** (1 - x)) * zeta(x)
    assert zeta(x).rewrite(dirichlet_eta) == dirichlet_eta(x) / (1 - 2 ** (1 - x))
    assert tn(dirichlet_eta(x), dirichlet_eta(x).rewrite(zeta), x)
    assert tn(zeta(x), zeta(x).rewrite(dirichlet_eta), x)

    assert zeta(x, a).rewrite(lerchphi) == lerchphi(1, x, a)
    assert polylog(s, z).rewrite(lerchphi) == lerchphi(z, s, 1) * z

    assert lerchphi(1, x, a).rewrite(zeta) == zeta(x, a)
    assert z * lerchphi(z, s, 1).rewrite(polylog) == polylog(s, z)
def test_polylog_values():
    from sympy.utilities.randtest import verify_numerically as tn
    assert polylog(2, 2) == pi**2 / 4 - I * pi * log(2)
    assert polylog(2, S.Half) == pi**2 / 12 - log(2)**2 / 2
    for z in [
            S.Half, 2, (sqrt(5) - 1) / 2, -(sqrt(5) - 1) / 2,
            -(sqrt(5) + 1) / 2, (3 - sqrt(5)) / 2
    ]:
        assert Abs(
            polylog(2, z).evalf() -
            polylog(2, z, evaluate=False).evalf()) < 1e-15
    z = Symbol("z")
    for s in [-1, 0]:
        for _ in range(10):
            assert tn(polylog(s, z),
                      polylog(s, z, evaluate=False),
                      z,
                      a=-3,
                      b=-2,
                      c=S.Half,
                      d=2)
            assert tn(polylog(s, z),
                      polylog(s, z, evaluate=False),
                      z,
                      a=2,
                      b=-2,
                      c=5,
                      d=2)
Ejemplo n.º 13
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def test_derivatives():
    from sympy import Derivative
    assert zeta(x, a).diff(x) == Derivative(zeta(x, a), x)
    assert zeta(x, a).diff(a) == -x*zeta(x + 1, a)
    assert lerchphi(z, s, a).diff(z) == (lerchphi(z, s-1, a) - a*lerchphi(z, s, a))/z
    assert lerchphi(z, s, a).diff(a) == -s*lerchphi(z, s+1, a)
    assert polylog(s, z).diff(z) == polylog(s - 1, z)/z

    b = randcplx()
    c = randcplx()
    assert td(zeta(b, x), x)
    assert td(polylog(b, z), z)
    assert td(lerchphi(c, b, x), x)
    assert td(lerchphi(x, b, c), x)
Ejemplo n.º 14
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def test_lerchphi():
    from sympy import combsimp, exp_polar, polylog, log, lerchphi
    assert hyperexpand(hyper([1, a], [a + 1], z)/a) == lerchphi(z, 1, a)
    assert hyperexpand(
        hyper([1, a, a], [a + 1, a + 1], z)/a**2) == lerchphi(z, 2, a)
    assert hyperexpand(hyper([1, a, a, a], [a + 1, a + 1, a + 1], z)/a**3) == \
           lerchphi(z, 3, a)
    assert hyperexpand(hyper([1] + [a]*10, [a + 1]*10, z)/a**10) \
           == lerchphi(z, 10, a)
    assert combsimp(hyperexpand(meijerg([0, 1 - a], [], [0], [-a],
                    exp_polar(-I*pi)*z))) == \
           lerchphi(z, 1, a)
    assert combsimp(hyperexpand(meijerg([0, 1 - a, 1 - a], [], [0], [-a, -a],
                    exp_polar(-I*pi)*z))) == \
           lerchphi(z, 2, a)
    assert combsimp(
        hyperexpand(meijerg([0, 1 - a, 1 - a, 1 - a], [], [0], [-a, -a, -a],
                    exp_polar(-I*pi)*z))) == \
           lerchphi(z, 3, a)

    assert hyperexpand(z*hyper([1, 1], [2], z)) == -log(1 + -z)
    assert hyperexpand(z*hyper([1, 1, 1], [2, 2], z)) == polylog(2, z)
    assert hyperexpand(z*hyper([1, 1, 1, 1], [2, 2, 2], z)) == polylog(3, z)

    assert hyperexpand(hyper([1, a, 1 + S(1)/2], [a + 1, S(1)/2], z)) == \
           -2*a/(z - 1) + (-2*a**2 + a)*lerchphi(z, 1, a)

    # Now numerical tests. These make sure reductions etc are carried out
    # correctly

    # a rational function (polylog at negative integer order)
    assert can_do([2, 2, 2], [1, 1])

    # NOTE these contain log(1-x) etc ... better make sure we have |z| < 1
    # reduction of order for polylog
    assert can_do([1, 1, 1, b + 5], [2, 2, b], div=10)

    # reduction of order for lerchphi
    # XXX lerchphi in mpmath is flaky
    assert can_do(
        [1, a, a, a, b + 5], [a + 1, a + 1, a + 1, b], numerical=False)

    # test a bug
    from sympy import Abs
    assert hyperexpand(hyper([S(1)/2, S(1)/2, S(1)/2, 1],
                             [S(3)/2, S(3)/2, S(3)/2], S(1)/4)) == \
        Abs(-polylog(3, exp_polar(I*pi)/2) + polylog(3, S(1)/2))
Ejemplo n.º 15
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def test_polylog2():
    x = polylog2(0.5)
    assert_allclose(x, 0.5822405264516294)
    
    from sympy import polylog
    xs = np.linspace(0,0.99999, 50)
    ys_act = [float(polylog(2, x)) for x in xs]
    ys = [polylog2(x) for x in xs]
    assert_allclose(ys, ys_act, rtol=1E-7, atol=1E-10)
Ejemplo n.º 16
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def test_polylog_expansion():
    from sympy import log
    assert polylog(s, 0) == 0
    assert polylog(s, 1) == zeta(s)
    assert polylog(s, -1) == -dirichlet_eta(s)

    assert myexpand(polylog(1, z), -log(1 - z))
    assert myexpand(polylog(0, z), z/(1 - z))
    assert myexpand(polylog(-1, z), z/(1 - z)**2)
    assert ((1-z)**3 * expand_func(polylog(-2, z))).simplify() == z*(1 + z)
    assert myexpand(polylog(-5, z), None)
Ejemplo n.º 17
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def test_derivatives():
    from sympy import Derivative
    assert zeta(x, a).diff(x) == Derivative(zeta(x, a), x)
    assert zeta(x, a).diff(a) == -x * zeta(x + 1, a)
    assert lerchphi(
        z, s, a).diff(z) == (lerchphi(z, s - 1, a) - a * lerchphi(z, s, a)) / z
    assert lerchphi(z, s, a).diff(a) == -s * lerchphi(z, s + 1, a)
    assert polylog(s, z).diff(z) == polylog(s - 1, z) / z

    b = randcplx()
    c = randcplx()
    assert td(zeta(b, x), x)
    assert td(polylog(b, z), z)
    assert td(lerchphi(c, b, x), x)
    assert td(lerchphi(x, b, c), x)
    raises(ArgumentIndexError, lambda: lerchphi(c, b, x).fdiff(2))
    raises(ArgumentIndexError, lambda: lerchphi(c, b, x).fdiff(4))
    raises(ArgumentIndexError, lambda: polylog(b, z).fdiff(1))
    raises(ArgumentIndexError, lambda: polylog(b, z).fdiff(3))
Ejemplo n.º 18
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def test_polylog_values():
    from sympy.utilities.randtest import verify_numerically as tn
    assert polylog(2, 2) == pi**2/4 - I*pi*log(2)
    assert polylog(2, S.Half) == pi**2/12 - log(2)**2/2
    for z in [S.Half, 2, (sqrt(5)-1)/2, -(sqrt(5)-1)/2, -(sqrt(5)+1)/2, (3-sqrt(5))/2]:
        assert Abs(polylog(2, z).evalf() - polylog(2, z, evaluate=False).evalf()) < 1e-15
    z = Symbol("z")
    for s in [-1, 0]:
        for _ in range(10):
            assert tn(polylog(s, z), polylog(s, z, evaluate=False), z,
                a=-3, b=-2, c=S.Half, d=2)
            assert tn(polylog(s, z), polylog(s, z, evaluate=False), z,
                a=2, b=-2, c=5, d=2)
Ejemplo n.º 19
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def test_polylog_expansion():
    from sympy import factor, log
    assert polylog(s, 0) == 0
    assert polylog(s, 1) == zeta(s)
    assert polylog(s, -1) == dirichlet_eta(s)

    assert myexpand(polylog(1, z), -log(1 + exp_polar(-I * pi) * z))
    assert myexpand(polylog(0, z), z / (1 - z))
    assert myexpand(polylog(-1, z), z**2 / (1 - z)**2 + z / (1 - z))
    assert myexpand(polylog(-5, z), None)
Ejemplo n.º 20
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def test_polylog_expansion():
    from sympy import factor, log
    assert polylog(s, 0) == 0
    assert polylog(s, 1) == zeta(s)
    assert polylog(s, -1) == dirichlet_eta(s)

    assert myexpand(polylog(1, z), -log(1 + exp_polar(-I*pi)*z))
    assert myexpand(polylog(0, z), z/(1 - z))
    assert myexpand(polylog(-1, z), z**2/(1 - z)**2 + z/(1 - z))
    assert myexpand(polylog(-5, z), None)
Ejemplo n.º 21
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def test_lerchphi_expansion():
    assert myexpand(lerchphi(1, s, a), zeta(s, a))
    assert myexpand(lerchphi(z, s, 1), polylog(s, z) / z)

    # direct summation
    assert myexpand(lerchphi(z, -1, a), a / (1 - z) + z / (1 - z) ** 2)
    assert myexpand(lerchphi(z, -3, a), None)
    # polylog reduction
    assert myexpand(
        lerchphi(z, s, S(1) / 2),
        2 ** (s - 1) * (polylog(s, sqrt(z)) / sqrt(z) - polylog(s, polar_lift(-1) * sqrt(z)) / sqrt(z)),
    )
    assert myexpand(lerchphi(z, s, 2), -1 / z + polylog(s, z) / z ** 2)
    assert myexpand(lerchphi(z, s, S(3) / 2), None)
    assert myexpand(lerchphi(z, s, S(7) / 3), None)
    assert myexpand(lerchphi(z, s, -S(1) / 3), None)
    assert myexpand(lerchphi(z, s, -S(5) / 2), None)

    # hurwitz zeta reduction
    assert myexpand(lerchphi(-1, s, a), 2 ** (-s) * zeta(s, a / 2) - 2 ** (-s) * zeta(s, (a + 1) / 2))
    assert myexpand(lerchphi(I, s, a), None)
    assert myexpand(lerchphi(-I, s, a), None)
    assert myexpand(lerchphi(exp(2 * I * pi / 5), s, a), None)
Ejemplo n.º 22
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def fdpolylog(eta):
    resultat=zeros(len(eta)); err=zeros(len(eta))
    for i in range(len(eta)):
        resultat[i] = - re(sqrt(pi)/2*polylog(2/3, -exp(eta[i]) ))
    return resultat, 0
Ejemplo n.º 23
0
def test_polylog_expansion():
    from sympy import log
    assert polylog(s, 0) == 0
    assert polylog(s, 1) == zeta(s)
    assert polylog(s, -1) == -dirichlet_eta(s)
    assert polylog(s, exp_polar(4*I*pi/3)) == polylog(s, exp(4*I*pi/3))
    assert polylog(s, exp_polar(I*pi)/3) == polylog(s, exp(I*pi)/3)

    assert myexpand(polylog(1, z), -log(1 - z))
    assert myexpand(polylog(0, z), z/(1 - z))
    assert myexpand(polylog(-1, z), z/(1 - z)**2)
    assert ((1-z)**3 * expand_func(polylog(-2, z))).simplify() == z*(1 + z)
    assert myexpand(polylog(-5, z), None)
Ejemplo n.º 24
0
def test_latex_functions():
    assert latex(exp(x)) == "e^{x}"
    assert latex(exp(1) + exp(2)) == "e + e^{2}"

    f = Function('f')
    assert latex(f(x)) == r'f{\left (x \right )}'
    assert latex(f) == r'f'

    g = Function('g')
    assert latex(g(x, y)) == r'g{\left (x,y \right )}'
    assert latex(g) == r'g'

    h = Function('h')
    assert latex(h(x, y, z)) == r'h{\left (x,y,z \right )}'
    assert latex(h) == r'h'

    Li = Function('Li')
    assert latex(Li) == r'\operatorname{Li}'
    assert latex(Li(x)) == r'\operatorname{Li}{\left (x \right )}'

    beta = Function('beta')

    # not to be confused with the beta function
    assert latex(beta(x)) == r"\beta{\left (x \right )}"
    assert latex(beta) == r"\beta"

    assert latex(sin(x)) == r"\sin{\left (x \right )}"
    assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
    assert latex(sin(2*x**2), fold_func_brackets=True) == \
        r"\sin {2 x^{2}}"
    assert latex(sin(x**2), fold_func_brackets=True) == \
        r"\sin {x^{2}}"

    assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
    assert latex(asin(x)**2, inv_trig_style="full") == \
        r"\arcsin^{2}{\left (x \right )}"
    assert latex(asin(x)**2, inv_trig_style="power") == \
        r"\sin^{-1}{\left (x \right )}^{2}"
    assert latex(asin(x**2), inv_trig_style="power",
                 fold_func_brackets=True) == \
        r"\sin^{-1} {x^{2}}"

    assert latex(factorial(k)) == r"k!"
    assert latex(factorial(-k)) == r"\left(- k\right)!"

    assert latex(subfactorial(k)) == r"!k"
    assert latex(subfactorial(-k)) == r"!\left(- k\right)"

    assert latex(factorial2(k)) == r"k!!"
    assert latex(factorial2(-k)) == r"\left(- k\right)!!"

    assert latex(binomial(2, k)) == r"{\binom{2}{k}}"

    assert latex(
        FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}"
    assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"

    assert latex(floor(x)) == r"\lfloor{x}\rfloor"
    assert latex(ceiling(x)) == r"\lceil{x}\rceil"
    assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
    assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
    assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
    assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
    assert latex(Abs(x)) == r"\left\lvert{x}\right\rvert"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(conjugate(x)) == r"\overline{x}"
    assert latex(gamma(x)) == r"\Gamma\left(x\right)"
    assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
    assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
    assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'

    assert latex(cot(x)) == r'\cot{\left (x \right )}'
    assert latex(coth(x)) == r'\coth{\left (x \right )}'
    assert latex(re(x)) == r'\Re{x}'
    assert latex(im(x)) == r'\Im{x}'
    assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
    assert latex(arg(x)) == r'\arg{\left (x \right )}'
    assert latex(zeta(x)) == r'\zeta\left(x\right)'

    assert latex(zeta(x)) == r"\zeta\left(x\right)"
    assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
    assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
    assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
    assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
    assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
    assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
    assert latex(
        polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
    assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
    assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"

    assert latex(elliptic_k(z)) == r"K\left(z\right)"
    assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
    assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
    assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
    assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
    assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
    assert latex(elliptic_e(z)) == r"E\left(z\right)"
    assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
    assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
    assert latex(elliptic_pi(x, y, z)**2) == \
        r"\Pi^{2}\left(x; y\middle| z\right)"
    assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
    assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"

    assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
    assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
    assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
    assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
    assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
    assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
    assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}', latex(Chi(x)**2)

    assert latex(
        jacobi(n, a, b, x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)'
    assert latex(jacobi(n, a, b, x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
    assert latex(
        gegenbauer(n, a, x)) == r'C_{n}^{\left(a\right)}\left(x\right)'
    assert latex(gegenbauer(n, a, x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
    assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
    assert latex(
        chebyshevt(n, x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}'
    assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
    assert latex(
        chebyshevu(n, x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}'
    assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
    assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
    assert latex(
        assoc_legendre(n, a, x)) == r'P_{n}^{\left(a\right)}\left(x\right)'
    assert latex(assoc_legendre(n, a, x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
    assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
    assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
    assert latex(
        assoc_laguerre(n, a, x)) == r'L_{n}^{\left(a\right)}\left(x\right)'
    assert latex(assoc_laguerre(n, a, x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
    assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
    assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'

    theta = Symbol("theta", real=True)
    phi = Symbol("phi", real=True)
    assert latex(Ynm(n,m,theta,phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
    assert latex(Ynm(n, m, theta, phi)**3) == r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
    assert latex(Znm(n,m,theta,phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
    assert latex(Znm(n, m, theta, phi)**3) == r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'

    # Test latex printing of function names with "_"
    assert latex(
        polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
    assert latex(polar_lift(
        0)**3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"

    assert latex(totient(n)) == r'\phi\left( n \right)'

    # some unknown function name should get rendered with \operatorname
    fjlkd = Function('fjlkd')
    assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left (x \right )}'
    # even when it is referred to without an argument
    assert latex(fjlkd) == r'\operatorname{fjlkd}'
Ejemplo n.º 25
0
def test_latex_functions():
    assert latex(exp(x)) == "e^{x}"
    assert latex(exp(1) + exp(2)) == "e + e^{2}"

    f = Function('f')
    assert latex(f(x)) == r'f{\left (x \right )}'
    assert latex(f) == r'f'

    g = Function('g')
    assert latex(g(x, y)) == r'g{\left (x,y \right )}'
    assert latex(g) == r'g'

    h = Function('h')
    assert latex(h(x, y, z)) == r'h{\left (x,y,z \right )}'
    assert latex(h) == r'h'

    Li = Function('Li')
    assert latex(Li) == r'\operatorname{Li}'
    assert latex(Li(x)) == r'\operatorname{Li}{\left (x \right )}'

    beta = Function('beta')

    # not to be confused with the beta function
    assert latex(beta(x)) == r"\beta{\left (x \right )}"
    assert latex(beta) == r"\beta"

    assert latex(sin(x)) == r"\sin{\left (x \right )}"
    assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
    assert latex(sin(2*x**2), fold_func_brackets=True) == \
        r"\sin {2 x^{2}}"
    assert latex(sin(x**2), fold_func_brackets=True) == \
        r"\sin {x^{2}}"

    assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
    assert latex(asin(x)**2, inv_trig_style="full") == \
        r"\arcsin^{2}{\left (x \right )}"
    assert latex(asin(x)**2, inv_trig_style="power") == \
        r"\sin^{-1}{\left (x \right )}^{2}"
    assert latex(asin(x**2), inv_trig_style="power",
                 fold_func_brackets=True) == \
        r"\sin^{-1} {x^{2}}"

    assert latex(factorial(k)) == r"k!"
    assert latex(factorial(-k)) == r"\left(- k\right)!"

    assert latex(subfactorial(k)) == r"!k"
    assert latex(subfactorial(-k)) == r"!\left(- k\right)"

    assert latex(factorial2(k)) == r"k!!"
    assert latex(factorial2(-k)) == r"\left(- k\right)!!"

    assert latex(binomial(2, k)) == r"{\binom{2}{k}}"

    assert latex(FallingFactorial(3,
                                  k)) == r"{\left(3\right)}_{\left(k\right)}"
    assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"

    assert latex(floor(x)) == r"\lfloor{x}\rfloor"
    assert latex(ceiling(x)) == r"\lceil{x}\rceil"
    assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
    assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
    assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
    assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
    assert latex(Abs(x)) == r"\left\lvert{x}\right\rvert"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(conjugate(x)) == r"\overline{x}"
    assert latex(gamma(x)) == r"\Gamma{\left(x \right)}"
    w = Wild('w')
    assert latex(gamma(w)) == r"\Gamma{\left(w \right)}"
    assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
    assert latex(Order(x, x)) == r"\mathcal{O}\left(x\right)"
    assert latex(Order(x, x, 0)) == r"\mathcal{O}\left(x\right)"
    assert latex(Order(x, x,
                       oo)) == r"\mathcal{O}\left(x; x\rightarrow\infty\right)"
    assert latex(
        Order(x, x, y)
    ) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow0\right)"
    assert latex(
        Order(x, x, y, 0)
    ) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow0\right)"
    assert latex(
        Order(x, x, y, oo)
    ) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow\infty\right)"
    assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
    assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'

    assert latex(cot(x)) == r'\cot{\left (x \right )}'
    assert latex(coth(x)) == r'\coth{\left (x \right )}'
    assert latex(re(x)) == r'\Re{x}'
    assert latex(im(x)) == r'\Im{x}'
    assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
    assert latex(arg(x)) == r'\arg{\left (x \right )}'
    assert latex(zeta(x)) == r'\zeta\left(x\right)'

    assert latex(zeta(x)) == r"\zeta\left(x\right)"
    assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
    assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
    assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
    assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
    assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
    assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
    assert latex(polylog(x,
                         y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
    assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
    assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"

    assert latex(elliptic_k(z)) == r"K\left(z\right)"
    assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
    assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
    assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
    assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
    assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
    assert latex(elliptic_e(z)) == r"E\left(z\right)"
    assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
    assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
    assert latex(elliptic_pi(x, y, z)**2) == \
        r"\Pi^{2}\left(x; y\middle| z\right)"
    assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
    assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"

    assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
    assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
    assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
    assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
    assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
    assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
    assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'
    assert latex(Chi(x)) == r'\operatorname{Chi}{\left (x \right )}'

    assert latex(jacobi(n, a, b,
                        x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)'
    assert latex(jacobi(
        n, a, b,
        x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
    assert latex(gegenbauer(n, a,
                            x)) == r'C_{n}^{\left(a\right)}\left(x\right)'
    assert latex(gegenbauer(
        n, a,
        x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
    assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
    assert latex(chebyshevt(n,
                            x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}'
    assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
    assert latex(chebyshevu(n,
                            x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}'
    assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
    assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
    assert latex(assoc_legendre(n, a,
                                x)) == r'P_{n}^{\left(a\right)}\left(x\right)'
    assert latex(assoc_legendre(
        n, a,
        x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
    assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
    assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
    assert latex(assoc_laguerre(n, a,
                                x)) == r'L_{n}^{\left(a\right)}\left(x\right)'
    assert latex(assoc_laguerre(
        n, a,
        x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
    assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
    assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'

    theta = Symbol("theta", real=True)
    phi = Symbol("phi", real=True)
    assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
    assert latex(
        Ynm(n, m, theta,
            phi)**3) == r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
    assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
    assert latex(
        Znm(n, m, theta,
            phi)**3) == r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'

    # Test latex printing of function names with "_"
    assert latex(
        polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
    assert latex(polar_lift(0)**
                 3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"

    assert latex(totient(n)) == r'\phi\left( n \right)'

    # some unknown function name should get rendered with \operatorname
    fjlkd = Function('fjlkd')
    assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left (x \right )}'
    # even when it is referred to without an argument
    assert latex(fjlkd) == r'\operatorname{fjlkd}'
Ejemplo n.º 26
0
def p_end_numerator_fea_approx(k, beta):
    return sum(
        polylog(beta - i, 1 / e) / factorial(i) for i in range(
            1, k + 1)) / normalization_constant_heavy_tailed_distribution(beta)
Ejemplo n.º 27
0
 def _characteristic_function(self, t):
     return polylog(self.s, exp(I * t)) / zeta(self.s)
Ejemplo n.º 28
0
def test_issue_10798():
    from sympy import integrate, pi, I, log, polylog, exp_polar
    from sympy.abc import x, y
    assert integrate(1/(1-(x*y)**2), (x, 0, 1), y) == \
        I*pi*log(y)/2 + polylog(2, exp_polar(I*pi)/y)/2 - \
        polylog(2, exp_polar(2*I*pi)/y)/2
Ejemplo n.º 29
0
def test_issue_10798():
    from sympy import integrate, pi, I, log, polylog, exp_polar
    from sympy.abc import x, y
    assert integrate(1/(1-(x*y)**2), (x, 0, 1), y) == \
        I*pi*log(y)/2 + polylog(2, exp_polar(I*pi)/y)/2 - \
        polylog(2, exp_polar(2*I*pi)/y)/2
Ejemplo n.º 30
0
 def _moment_generating_function(self, t):
     return polylog(self.s, exp(t)) / zeta(self.s)
Ejemplo n.º 31
0
def test_polylog_expansion():
    assert polylog(s, 0) == 0
    assert polylog(s, 1) == zeta(s)
    assert polylog(s, -1) == -dirichlet_eta(s)
    assert polylog(s, exp_polar(I * pi * Rational(4, 3))) == polylog(
        s, exp(I * pi * Rational(4, 3)))
    assert polylog(s, exp_polar(I * pi) / 3) == polylog(s, exp(I * pi) / 3)

    assert myexpand(polylog(1, z), -log(1 - z))
    assert myexpand(polylog(0, z), z / (1 - z))
    assert myexpand(polylog(-1, z), z / (1 - z)**2)
    assert ((1 - z)**3 * expand_func(polylog(-2, z))).simplify() == z * (1 + z)
    assert myexpand(polylog(-5, z), None)
Ejemplo n.º 32
0
def test_latex_functions():
    assert latex(exp(x)) == "e^{x}"
    assert latex(exp(1) + exp(2)) == "e + e^{2}"

    f = Function('f')
    assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'

    beta = Function('beta')

    assert latex(beta(x)) == r"\beta{\left (x \right )}"
    assert latex(sin(x)) == r"\sin{\left (x \right )}"
    assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
    assert latex(sin(2*x**2), fold_func_brackets=True) == \
        r"\sin {2 x^{2}}"
    assert latex(sin(x**2), fold_func_brackets=True) == \
        r"\sin {x^{2}}"

    assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
    assert latex(asin(x)**2, inv_trig_style="full") == \
        r"\arcsin^{2}{\left (x \right )}"
    assert latex(asin(x)**2, inv_trig_style="power") == \
        r"\sin^{-1}{\left (x \right )}^{2}"
    assert latex(asin(x**2), inv_trig_style="power",
                 fold_func_brackets=True) == \
        r"\sin^{-1} {x^{2}}"

    assert latex(factorial(k)) == r"k!"
    assert latex(factorial(-k)) == r"\left(- k\right)!"

    assert latex(factorial2(k)) == r"k!!"
    assert latex(factorial2(-k)) == r"\left(- k\right)!!"

    assert latex(binomial(2, k)) == r"{\binom{2}{k}}"

    assert latex(FallingFactorial(3,
                                  k)) == r"{\left(3\right)}_{\left(k\right)}"
    assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"

    assert latex(floor(x)) == r"\lfloor{x}\rfloor"
    assert latex(ceiling(x)) == r"\lceil{x}\rceil"
    assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
    assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
    assert latex(Abs(x)) == r"\lvert{x}\rvert"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(re(x + y)) == r"\Re {\left (x + y \right )}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(conjugate(x)) == r"\overline{x}"
    assert latex(gamma(x)) == r"\Gamma\left(x\right)"
    assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
    assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
    assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'

    assert latex(cot(x)) == r'\cot{\left (x \right )}'
    assert latex(coth(x)) == r'\coth{\left (x \right )}'
    assert latex(re(x)) == r'\Re{x}'
    assert latex(im(x)) == r'\Im{x}'
    assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
    assert latex(arg(x)) == r'\arg{\left (x \right )}'
    assert latex(zeta(x)) == r'\zeta\left(x\right)'

    assert latex(zeta(x)) == r"\zeta\left(x\right)"
    assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
    assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
    assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
    assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
    assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
    assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
    assert latex(polylog(x,
                         y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
    assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
    assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"

    assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
    assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
    assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
    assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
    assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
    assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
    assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'
Ejemplo n.º 33
0
 def _characteristic_function(self, t):
     return polylog(self.s, exp(I*t)) / zeta(self.s)
Ejemplo n.º 34
0
def test_latex_functions():
    assert latex(exp(x)) == "e^{x}"
    assert latex(exp(1) + exp(2)) == "e + e^{2}"

    f = Function("f")
    assert latex(f(x)) == "\\operatorname{f}{\\left (x \\right )}"

    beta = Function("beta")

    assert latex(beta(x)) == r"\beta{\left (x \right )}"
    assert latex(sin(x)) == r"\sin{\left (x \right )}"
    assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
    assert latex(sin(2 * x ** 2), fold_func_brackets=True) == r"\sin {2 x^{2}}"
    assert latex(sin(x ** 2), fold_func_brackets=True) == r"\sin {x^{2}}"

    assert latex(asin(x) ** 2) == r"\operatorname{asin}^{2}{\left (x \right )}"
    assert latex(asin(x) ** 2, inv_trig_style="full") == r"\arcsin^{2}{\left (x \right )}"
    assert latex(asin(x) ** 2, inv_trig_style="power") == r"\sin^{-1}{\left (x \right )}^{2}"
    assert latex(asin(x ** 2), inv_trig_style="power", fold_func_brackets=True) == r"\sin^{-1} {x^{2}}"

    assert latex(factorial(k)) == r"k!"
    assert latex(factorial(-k)) == r"\left(- k\right)!"

    assert latex(subfactorial(k)) == r"!k"
    assert latex(subfactorial(-k)) == r"!\left(- k\right)"

    assert latex(factorial2(k)) == r"k!!"
    assert latex(factorial2(-k)) == r"\left(- k\right)!!"

    assert latex(binomial(2, k)) == r"{\binom{2}{k}}"

    assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}"
    assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"

    assert latex(floor(x)) == r"\lfloor{x}\rfloor"
    assert latex(ceiling(x)) == r"\lceil{x}\rceil"
    assert latex(Min(x, 2, x ** 3)) == r"\min\left(2, x, x^{3}\right)"
    assert latex(Min(x, y) ** 2) == r"\min\left(x, y\right)^{2}"
    assert latex(Max(x, 2, x ** 3)) == r"\max\left(2, x, x^{3}\right)"
    assert latex(Max(x, y) ** 2) == r"\max\left(x, y\right)^{2}"
    assert latex(Abs(x)) == r"\lvert{x}\rvert"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(conjugate(x)) == r"\overline{x}"
    assert latex(gamma(x)) == r"\Gamma\left(x\right)"
    assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
    assert latex(lowergamma(x, y)) == r"\gamma\left(x, y\right)"
    assert latex(uppergamma(x, y)) == r"\Gamma\left(x, y\right)"

    assert latex(cot(x)) == r"\cot{\left (x \right )}"
    assert latex(coth(x)) == r"\coth{\left (x \right )}"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(root(x, y)) == r"x^{\frac{1}{y}}"
    assert latex(arg(x)) == r"\arg{\left (x \right )}"
    assert latex(zeta(x)) == r"\zeta\left(x\right)"

    assert latex(zeta(x)) == r"\zeta\left(x\right)"
    assert latex(zeta(x) ** 2) == r"\zeta^{2}\left(x\right)"
    assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
    assert latex(zeta(x, y) ** 2) == r"\zeta^{2}\left(x, y\right)"
    assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
    assert latex(dirichlet_eta(x) ** 2) == r"\eta^{2}\left(x\right)"
    assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
    assert latex(polylog(x, y) ** 2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
    assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
    assert latex(lerchphi(x, y, n) ** 2) == r"\Phi^{2}\left(x, y, n\right)"

    assert latex(Ei(x)) == r"\operatorname{Ei}{\left (x \right )}"
    assert latex(Ei(x) ** 2) == r"\operatorname{Ei}^{2}{\left (x \right )}"
    assert latex(expint(x, y) ** 2) == r"\operatorname{E}_{x}^{2}\left(y\right)"
    assert latex(Shi(x) ** 2) == r"\operatorname{Shi}^{2}{\left (x \right )}"
    assert latex(Si(x) ** 2) == r"\operatorname{Si}^{2}{\left (x \right )}"
    assert latex(Ci(x) ** 2) == r"\operatorname{Ci}^{2}{\left (x \right )}"
    assert latex(Chi(x) ** 2) == r"\operatorname{Chi}^{2}{\left (x \right )}"

    assert latex(jacobi(n, a, b, x)) == r"P_{n}^{\left(a,b\right)}\left(x\right)"
    assert latex(jacobi(n, a, b, x) ** 2) == r"\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}"
    assert latex(gegenbauer(n, a, x)) == r"C_{n}^{\left(a\right)}\left(x\right)"
    assert latex(gegenbauer(n, a, x) ** 2) == r"\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}"
    assert latex(chebyshevt(n, x)) == r"T_{n}\left(x\right)"
    assert latex(chebyshevt(n, x) ** 2) == r"\left(T_{n}\left(x\right)\right)^{2}"
    assert latex(chebyshevu(n, x)) == r"U_{n}\left(x\right)"
    assert latex(chebyshevu(n, x) ** 2) == r"\left(U_{n}\left(x\right)\right)^{2}"
    assert latex(legendre(n, x)) == r"P_{n}\left(x\right)"
    assert latex(legendre(n, x) ** 2) == r"\left(P_{n}\left(x\right)\right)^{2}"
    assert latex(assoc_legendre(n, a, x)) == r"P_{n}^{\left(a\right)}\left(x\right)"
    assert latex(assoc_legendre(n, a, x) ** 2) == r"\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}"
    assert latex(laguerre(n, x)) == r"L_{n}\left(x\right)"
    assert latex(laguerre(n, x) ** 2) == r"\left(L_{n}\left(x\right)\right)^{2}"
    assert latex(assoc_laguerre(n, a, x)) == r"L_{n}^{\left(a\right)}\left(x\right)"
    assert latex(assoc_laguerre(n, a, x) ** 2) == r"\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}"
    assert latex(hermite(n, x)) == r"H_{n}\left(x\right)"
    assert latex(hermite(n, x) ** 2) == r"\left(H_{n}\left(x\right)\right)^{2}"

    # Test latex printing of function names with "_"
    assert latex(polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
    assert latex(polar_lift(0) ** 3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
Ejemplo n.º 35
0
def test_latex_functions():
    assert latex(exp(x)) == "e^{x}"
    assert latex(exp(1)+exp(2)) == "e + e^{2}"

    f = Function('f')
    assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'

    beta = Function('beta')

    assert latex(beta(x)) == r"\beta{\left (x \right )}"
    assert latex(sin(x)) == r"\sin{\left (x \right )}"
    assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
    assert latex(sin(2*x**2), fold_func_brackets=True) == \
    r"\sin {2 x^{2}}"
    assert latex(sin(x**2), fold_func_brackets=True) == \
    r"\sin {x^{2}}"

    assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
    assert latex(asin(x)**2,inv_trig_style="full") == \
        r"\arcsin^{2}{\left (x \right )}"
    assert latex(asin(x)**2,inv_trig_style="power") == \
        r"\sin^{-1}{\left (x \right )}^{2}"
    assert latex(asin(x**2),inv_trig_style="power",fold_func_brackets=True) == \
        r"\sin^{-1} {x^{2}}"

    assert latex(factorial(k)) == r"k!"
    assert latex(factorial(-k)) == r"\left(- k\right)!"

    assert latex(factorial2(k)) == r"k!!"
    assert latex(factorial2(-k)) == r"\left(- k\right)!!"

    assert latex(binomial(2,k)) == r"{\binom{2}{k}}"

    assert latex(FallingFactorial(3,k)) == r"{\left(3\right)}_{\left(k\right)}"
    assert latex(RisingFactorial(3,k)) == r"{\left(3\right)}^{\left(k\right)}"

    assert latex(floor(x)) == r"\lfloor{x}\rfloor"
    assert latex(ceiling(x)) == r"\lceil{x}\rceil"
    assert latex(Abs(x)) == r"\lvert{x}\rvert"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(re(x+y)) == r"\Re {\left (x + y \right )}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(conjugate(x)) == r"\overline{x}"
    assert latex(gamma(x)) == r"\Gamma\left(x\right)"
    assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
    assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
    assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'

    assert latex(cot(x)) == r'\cot{\left (x \right )}'
    assert latex(coth(x)) == r'\coth{\left (x \right )}'
    assert latex(re(x)) == r'\Re{x}'
    assert latex(im(x)) == r'\Im{x}'
    assert latex(root(x,y)) == r'x^{\frac{1}{y}}'
    assert latex(arg(x)) == r'\arg{\left (x \right )}'
    assert latex(zeta(x)) == r'\zeta\left(x\right)'

    assert latex(zeta(x)) == r"\zeta\left(x\right)"
    assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
    assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
    assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
    assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
    assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
    assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
    assert latex(polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
    assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
    assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"

    assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
    assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
    assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
    assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
    assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
    assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
    assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'
Ejemplo n.º 36
0
 def _moment_generating_function(self, t):
     return polylog(self.s, exp(t)) / zeta(self.s)
Ejemplo n.º 37
0
 def cbx(self, nmax):
     self.cb[0] = self.pb
     for i in range(2, nmax):
         self.cb[i - 1] = simplify(
             expand_func(polylog(1 - i, 1 - 1 / self.pb)))
Ejemplo n.º 38
0
def test_latex_functions():
    assert latex(exp(x)) == "e^{x}"
    assert latex(exp(1) + exp(2)) == "e + e^{2}"

    f = Function('f')
    assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'

    beta = Function('beta')

    assert latex(beta(x)) == r"\beta{\left (x \right )}"
    assert latex(sin(x)) == r"\sin{\left (x \right )}"
    assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
    assert latex(sin(2*x**2), fold_func_brackets=True) == \
        r"\sin {2 x^{2}}"
    assert latex(sin(x**2), fold_func_brackets=True) == \
        r"\sin {x^{2}}"

    assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
    assert latex(asin(x)**2, inv_trig_style="full") == \
        r"\arcsin^{2}{\left (x \right )}"
    assert latex(asin(x)**2, inv_trig_style="power") == \
        r"\sin^{-1}{\left (x \right )}^{2}"
    assert latex(asin(x**2), inv_trig_style="power",
                 fold_func_brackets=True) == \
        r"\sin^{-1} {x^{2}}"

    assert latex(factorial(k)) == r"k!"
    assert latex(factorial(-k)) == r"\left(- k\right)!"

    assert latex(subfactorial(k)) == r"!k"
    assert latex(subfactorial(-k)) == r"!\left(- k\right)"

    assert latex(factorial2(k)) == r"k!!"
    assert latex(factorial2(-k)) == r"\left(- k\right)!!"

    assert latex(binomial(2, k)) == r"{\binom{2}{k}}"

    assert latex(FallingFactorial(3,
                                  k)) == r"{\left(3\right)}_{\left(k\right)}"
    assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"

    assert latex(floor(x)) == r"\lfloor{x}\rfloor"
    assert latex(ceiling(x)) == r"\lceil{x}\rceil"
    assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
    assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
    assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
    assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
    assert latex(Abs(x)) == r"\lvert{x}\rvert"
    assert latex(re(x)) == r"\Re{x}"
    assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
    assert latex(im(x)) == r"\Im{x}"
    assert latex(conjugate(x)) == r"\overline{x}"
    assert latex(gamma(x)) == r"\Gamma\left(x\right)"
    assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
    assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
    assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'

    assert latex(cot(x)) == r'\cot{\left (x \right )}'
    assert latex(coth(x)) == r'\coth{\left (x \right )}'
    assert latex(re(x)) == r'\Re{x}'
    assert latex(im(x)) == r'\Im{x}'
    assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
    assert latex(arg(x)) == r'\arg{\left (x \right )}'
    assert latex(zeta(x)) == r'\zeta\left(x\right)'

    assert latex(zeta(x)) == r"\zeta\left(x\right)"
    assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
    assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
    assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
    assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
    assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
    assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
    assert latex(polylog(x,
                         y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
    assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
    assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"

    assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
    assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
    assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
    assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
    assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
    assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
    assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'

    assert latex(jacobi(n, a, b,
                        x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)'
    assert latex(jacobi(
        n, a, b,
        x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
    assert latex(gegenbauer(n, a,
                            x)) == r'C_{n}^{\left(a\right)}\left(x\right)'
    assert latex(gegenbauer(
        n, a,
        x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
    assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
    assert latex(chebyshevt(n,
                            x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}'
    assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
    assert latex(chebyshevu(n,
                            x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}'
    assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
    assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
    assert latex(assoc_legendre(n, a,
                                x)) == r'P_{n}^{\left(a\right)}\left(x\right)'
    assert latex(assoc_legendre(
        n, a,
        x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
    assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
    assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
    assert latex(assoc_laguerre(n, a,
                                x)) == r'L_{n}^{\left(a\right)}\left(x\right)'
    assert latex(assoc_laguerre(
        n, a,
        x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
    assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
    assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'

    # Test latex printing of function names with "_"
    assert latex(
        polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
    assert latex(polar_lift(0)**
                 3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
Ejemplo n.º 39
0
 def cx(self, nmax):
     self.c[0] = self.p
     for i in range(2, nmax):
         self.c[i - 1] = (-1)**(i - 1) * simplify(
             expand_func(polylog(1 - i, 1 - 1 / self.p)))