def test_erf2():

    assert erf2(0, 0) == S.Zero
    assert erf2(x, x) == S.Zero
    assert erf2(nan, 0) == nan

    assert erf2(-oo,  y) ==  erf(y) + 1
    assert erf2( oo,  y) ==  erf(y) - 1
    assert erf2(  x, oo) ==  1 - erf(x)
    assert erf2(  x,-oo) == -1 - erf(x)
    assert erf2(x, erf2inv(x, y)) == y

    assert erf2(-x, -y) == -erf2(x,y)
    assert erf2(-x,  y) == erf(y) + erf(x)
    assert erf2( x, -y) == -erf(y) - erf(x)
    assert erf2(x, y).rewrite('fresnels') == erf(y).rewrite(fresnels)-erf(x).rewrite(fresnels)
    assert erf2(x, y).rewrite('fresnelc') == erf(y).rewrite(fresnelc)-erf(x).rewrite(fresnelc)
    assert erf2(x, y).rewrite('hyper') == erf(y).rewrite(hyper)-erf(x).rewrite(hyper)
    assert erf2(x, y).rewrite('meijerg') == erf(y).rewrite(meijerg)-erf(x).rewrite(meijerg)
    assert erf2(x, y).rewrite('uppergamma') == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma)
    assert erf2(x, y).rewrite('expint') == erf(y).rewrite(expint)-erf(x).rewrite(expint)

    assert erf2(I, 0).is_real is False
    assert erf2(0, 0).is_real is True

    assert expand_func(erf(x) + erf2(x, y)) == erf(y)

    assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y))

    assert erf2(x, y).rewrite('erf')  == erf(y) - erf(x)
    assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y)
    assert erf2(x, y).rewrite('erfi') == I*(erfi(I*x) - erfi(I*y))

    raises(ArgumentIndexError, lambda: erfi(x).fdiff(3))

def test_errorinverses():
    assert solveset_real(erf(x) - S.One/2, x) == \
        FiniteSet(erfinv(S.One/2))
    assert solveset_real(erfinv(x) - 2, x) == \
        FiniteSet(erf(2))
    assert solveset_real(erfc(x) - S.One, x) == \
        FiniteSet(erfcinv(S.One))
    assert solveset_real(erfcinv(x) - 2, x) == FiniteSet(erfc(2))

def test_erf():
    assert erf(nan) == nan

    assert erf(oo) == 1
    assert erf(-oo) == -1

    assert erf(0) == 0

    assert erf(I*oo) == oo*I
    assert erf(-I*oo) == -oo*I

    assert erf(-2) == -erf(2)
    assert erf(-x*y) == -erf(x*y)
    assert erf(-x - y) == -erf(x + y)

    assert erf(erfinv(x)) == x
    assert erf(erfcinv(x)) == 1 - x
    assert erf(erf2inv(0, x)) == x
    assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x

    assert erf(I).is_real is False
    assert erf(0).is_real is True

    assert conjugate(erf(z)) == erf(conjugate(z))

    assert erf(x).as_leading_term(x) == 2*x/sqrt(pi)
    assert erf(1/x).as_leading_term(x) == erf(1/x)

    assert erf(z).rewrite('uppergamma') == sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z
    assert erf(z).rewrite('erfc') == S.One - erfc(z)
    assert erf(z).rewrite('erfi') == -I*erfi(I*z)
    assert erf(z).rewrite('fresnels') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
        I*fresnels(z*(1 - I)/sqrt(pi)))
    assert erf(z).rewrite('fresnelc') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
        I*fresnels(z*(1 - I)/sqrt(pi)))
    assert erf(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi)
    assert erf(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [-S.Half], z**2)/sqrt(pi)
    assert erf(z).rewrite('expint') == sqrt(z**2)/z - z*expint(S.Half, z**2)/sqrt(S.Pi)

    assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \
        2/sqrt(pi)
    assert limit((1 - erf(z))*exp(z**2)*z, z, oo) == 1/sqrt(pi)
    assert limit((1 - erf(x))*exp(x**2)*sqrt(pi)*x, x, oo) == 1
    assert limit(((1 - erf(x))*exp(x**2)*sqrt(pi)*x - 1)*2*x**2, x, oo) == -1

    assert erf(x).as_real_imag() == \
        ((erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
         erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
         I*(erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
         erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
         re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))

    raises(ArgumentIndexError, lambda: erf(x).fdiff(2))

def test_to_hyper():
    x = symbols('x')
    R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
    p = HolonomicFunction(Dx - 2, x, 0, [3]).to_hyper()
    q = 3 * hyper([], [], 2*x)
    assert p == q
    p = hyperexpand(HolonomicFunction((1 + x) * Dx - 3, x, 0, [2]).to_hyper()).expand()
    q = 2*x**3 + 6*x**2 + 6*x + 2
    assert p == q
    p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]).to_hyper()
    q = -x**2*hyper((2, 2, 1), (2, 3), -x)/2 + x
    assert p == q
    p = HolonomicFunction(2*x*Dx + Dx**2, x, 0, [0, 2/sqrt(pi)]).to_hyper()
    q = 2*x*hyper((1/2,), (3/2,), -x**2)/sqrt(pi)
    assert p == q
    p = hyperexpand(HolonomicFunction(2*x*Dx + Dx**2, x, 0, [1, -2/sqrt(pi)]).to_hyper())
    q = erfc(x)
    assert p.rewrite(erfc) == q
    p =  hyperexpand(HolonomicFunction((x**2 - 1) + x*Dx + x**2*Dx**2,
        x, 0, [0, S(1)/2]).to_hyper())
    q = besselj(1, x)
    assert p == q
    p = hyperexpand(HolonomicFunction(x*Dx**2 + Dx + x, x, 0, [1, 0]).to_hyper())
    q = besselj(0, x)
    assert p == q

def test_hyper():
    for x in sorted(exparg):
        test("erf", x, N(sp.erf(x)))
    for x in sorted(exparg):
        test("erfc", x, N(sp.erfc(x)))

    gamarg = FiniteSet(*(x+S(1)/12 for x in exparg))
    betarg = ProductSet(gamarg, gamarg)
    for x in sorted(gamarg):
        test("lgamma", x, N(sp.log(abs(sp.gamma(x)))))
    for x in sorted(gamarg):
        test("gamma", x, N(sp.gamma(x)))
    for x, y in sorted(betarg, key=lambda (x, y): (y, x)):
        test("beta", x, y, N(sp.beta(x, y)))

    pgamarg = FiniteSet(S(1)/12, S(1)/3, S(3)/2, 5)
    pgamargp = ProductSet(gamarg & Interval(0, oo, True), pgamarg)
    for a, x in sorted(pgamargp):
        test("pgamma", a, x, N(sp.lowergamma(a, x)))
    for a, x in sorted(pgamargp):
        test("pgammac", a, x, N(sp.uppergamma(a, x)))
    for a, x in sorted(pgamargp):
        test("pgammar", a, x, N(sp.lowergamma(a, x)/sp.gamma(a)))
    for a, x in sorted(pgamargp):
        test("pgammarc", a, x, N(sp.uppergamma(a, x)/sp.gamma(a)))
    for a, x in sorted(pgamargp):
        test("ipgammarc", a, N(sp.uppergamma(a, x)/sp.gamma(a)), x)

    pbetargp = [(a, b, x) for a, b, x in ProductSet(betarg, pgamarg)
                if a > 0 and b > 0 and x < 1]
    pbetargp.sort(key=lambda (a, b, x): (b, a, x))
    for a, b, x in pbetargp:
        test("pbeta", a, b, x, mp.betainc(mpf(a), mpf(b), x2=mpf(x)))
    for a, b, x in pbetargp:
        test("pbetar", a, b, x, mp.betainc(mpf(a), mpf(b), x2=mpf(x),
                                           regularized=True))
    for a, b, x in pbetargp:
        test("ipbetar", a, b, mp.betainc(mpf(a), mpf(b), x2=mpf(x),
                                         regularized=True), x)

    for x in sorted(posarg):
        test("j0", x, N(sp.besselj(0, x)))
    for x in sorted(posarg):
        test("j1", x, N(sp.besselj(1, x)))
    for x in sorted(posarg-FiniteSet(0)):
        test("y0", x, N(sp.bessely(0, x)))
    for x in sorted(posarg-FiniteSet(0)):
        test("y1", x, N(sp.bessely(1, x)))

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)*erfc(sqrt(x))/(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)

    assert expint(S(3)/2, z).nseries(z) == \
        2 + 2*z - z**2/3 + z**3/15 - z**4/84 + z**5/540 - \
        2*sqrt(pi)*sqrt(z) + O(z**6)

    assert E1(z).series(z) == -EulerGamma - log(z) + z - \
        z**2/4 + z**3/18 - z**4/96 + z**5/600 + O(z**6)

    assert expint(4, z).series(z) == S(1)/3 - z/2 + z**2/2 + \
        z**3*(log(z)/6 - S(11)/36 + EulerGamma/6) - z**4/24 + \
        z**5/240 + O(z**6)

def test_erfi():
    assert erfi(nan) == nan

    assert erfi(oo) == S.Infinity
    assert erfi(-oo) == S.NegativeInfinity

    assert erfi(0) == S.Zero

    assert erfi(I*oo) == I
    assert erfi(-I*oo) == -I

    assert erfi(-x) == -erfi(x)

    assert erfi(I*erfinv(x)) == I*x
    assert erfi(I*erfcinv(x)) == I*(1 - x)
    assert erfi(I*erf2inv(0, x)) == I*x

    assert erfi(I).is_real is False
    assert erfi(0).is_real is True

    assert conjugate(erfi(z)) == erfi(conjugate(z))

    assert erfi(z).rewrite('erf') == -I*erf(I*z)
    assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I
    assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
        I*fresnels(z*(1 + I)/sqrt(pi)))
    assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
        I*fresnels(z*(1 + I)/sqrt(pi)))
    assert erfi(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], z**2)/sqrt(pi)
    assert erfi(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [-S.Half], -z**2)/sqrt(pi)
    assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(S.Half,
        -z**2)/sqrt(S.Pi) - S.One))
    assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi)
    assert expand_func(erfi(I*z)) == I*erf(z)

    assert erfi(x).as_real_imag() == \
        ((erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
         erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
         I*(erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
         erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
         re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))

    raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))

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)*erfc(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 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 conjugate(uppergamma(x, y)) == uppergamma(conjugate(x), conjugate(y))
    assert conjugate(uppergamma(x, 0)) == gamma(conjugate(x))
    assert conjugate(uppergamma(x, -oo)) == conjugate(uppergamma(x, -oo))

    assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y)
    assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)

    assert uppergamma(70, 6) == 69035724522603011058660187038367026272747334489677105069435923032634389419656200387949342530805432320*exp(-6)
    assert (uppergamma(S(77) / 2, 6) - uppergamma(S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
    assert (uppergamma(-S(77) / 2, 6) - uppergamma(-S(77) / 2, 6, evaluate=False)).evalf() < 1e-16

def _create_table(table):
    """
    Creates the look-up table. For a similar implementation
    see meijerint._create_lookup_table.
    """

    def add(formula, annihilator, arg, x0=0, y0=[]):
        """
        Adds a formula in the dictionary
        """
        table.setdefault(_mytype(formula, x_1), []).append((formula,
            HolonomicFunction(annihilator, arg, x0, y0)))

    R = QQ.old_poly_ring(x_1)
    _, Dx = DifferentialOperators(R, 'Dx')

    from sympy import (sin, cos, exp, log, erf, sqrt, pi,
        sinh, cosh, sinc, erfc, Si, Ci, Shi, erfi)

    # add some basic functions
    add(sin(x_1), Dx**2 + 1, x_1, 0, [0, 1])
    add(cos(x_1), Dx**2 + 1, x_1, 0, [1, 0])
    add(exp(x_1), Dx - 1, x_1, 0, 1)
    add(log(x_1), Dx + x_1*Dx**2, x_1, 1, [0, 1])

    add(erf(x_1), 2*x_1*Dx + Dx**2, x_1, 0, [0, 2/sqrt(pi)])
    add(erfc(x_1), 2*x_1*Dx + Dx**2, x_1, 0, [1, -2/sqrt(pi)])
    add(erfi(x_1), -2*x_1*Dx + Dx**2, x_1, 0, [0, 2/sqrt(pi)])

    add(sinh(x_1), Dx**2 - 1, x_1, 0, [0, 1])
    add(cosh(x_1), Dx**2 - 1, x_1, 0, [1, 0])

    add(sinc(x_1), x_1 + 2*Dx + x_1*Dx**2, x_1)

    add(Si(x_1), x_1*Dx + 2*Dx**2 + x_1*Dx**3, x_1)
    add(Ci(x_1), x_1*Dx + 2*Dx**2 + x_1*Dx**3, x_1)

    add(Shi(x_1), -x_1*Dx + 2*Dx**2 + x_1*Dx**3, x_1)

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)*erfc(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 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 conjugate(uppergamma(x, y)) == uppergamma(conjugate(x), conjugate(y))
    assert conjugate(uppergamma(x, 0)) == gamma(conjugate(x))
    assert conjugate(uppergamma(x, -oo)) == conjugate(uppergamma(x, -oo))

    assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y)
    assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)

def test_erfc():
    assert erfc(nan) == nan

    assert erfc(oo) == 0
    assert erfc(-oo) == 2

    assert erfc(0) == 1

    assert erfc(I*oo) == -oo*I
    assert erfc(-I*oo) == oo*I

    assert erfc(-x) == S(2) - erfc(x)
    assert erfc(erfcinv(x)) == x

    assert erfc(I).is_real is False
    assert erfc(0).is_real is True

    assert conjugate(erfc(z)) == erfc(conjugate(z))

    assert erfc(x).as_leading_term(x) == S.One
    assert erfc(1/x).as_leading_term(x) == erfc(1/x)

    assert erfc(z).rewrite('erf') == 1 - erf(z)
    assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z)
    assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
        I*fresnels(z*(1 - I)/sqrt(pi)))
    assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
        I*fresnels(z*(1 - I)/sqrt(pi)))
    assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi)
    assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [-S.Half], z**2)/sqrt(pi)
    assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z
    assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi)
    assert expand_func(erf(x) + erfc(x)) == S.One

    assert erfc(x).as_real_imag() == \
        ((erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
         erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
         I*(erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
         erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
         re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))

    raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))

def test_laplace_transform():
    from sympy import fresnels, fresnelc

    LT = laplace_transform
    a, b, c, = symbols("a b c", positive=True)
    t = symbols("t")
    w = Symbol("w")
    f = Function("f")

    # Test unevaluated form
    assert laplace_transform(f(t), t, w) == LaplaceTransform(f(t), t, w)
    assert inverse_laplace_transform(f(w), w, t, plane=0) == InverseLaplaceTransform(f(w), w, t, 0)

    # test a bug
    spos = symbols("s", positive=True)
    assert LT(exp(t), t, spos)[:2] == (1 / (spos - 1), True)

    # basic tests from wikipedia

    assert LT((t - a) ** b * exp(-c * (t - a)) * Heaviside(t - a), t, s) == (
        (s + c) ** (-b - 1) * exp(-a * s) * gamma(b + 1),
        -c,
        True,
    )
    assert LT(t ** a, t, s) == (s ** (-a - 1) * gamma(a + 1), 0, True)
    assert LT(Heaviside(t), t, s) == (1 / s, 0, True)
    assert LT(Heaviside(t - a), t, s) == (exp(-a * s) / s, 0, True)
    assert LT(1 - exp(-a * t), t, s) == (a / (s * (a + s)), 0, True)

    assert LT((exp(2 * t) - 1) * exp(-b - t) * Heaviside(t) / 2, t, s, noconds=True) == exp(-b) / (s ** 2 - 1)

    assert LT(exp(t), t, s)[:2] == (1 / (s - 1), 1)
    assert LT(exp(2 * t), t, s)[:2] == (1 / (s - 2), 2)
    assert LT(exp(a * t), t, s)[:2] == (1 / (s - a), a)

    assert LT(log(t / a), t, s) == ((log(a * s) + EulerGamma) / s / -1, 0, True)

    assert LT(erf(t), t, s) == ((erfc(s / 2)) * exp(s ** 2 / 4) / s, 0, True)

    assert LT(sin(a * t), t, s) == (a / (a ** 2 + s ** 2), 0, True)
    assert LT(cos(a * t), t, s) == (s / (a ** 2 + s ** 2), 0, True)
    # TODO would be nice to have these come out better
    assert LT(exp(-a * t) * sin(b * t), t, s) == (b / (b ** 2 + (a + s) ** 2), -a, True)
    assert LT(exp(-a * t) * cos(b * t), t, s) == ((a + s) / (b ** 2 + (a + s) ** 2), -a, True)

    assert LT(besselj(0, t), t, s) == (1 / sqrt(1 + s ** 2), 0, True)
    assert LT(besselj(1, t), t, s) == (1 - 1 / sqrt(1 + 1 / s ** 2), 0, True)
    # TODO general order works, but is a *mess*
    # TODO besseli also works, but is an even greater mess

    # test a bug in conditions processing
    # TODO the auxiliary condition should be recognised/simplified
    assert LT(exp(t) * cos(t), t, s)[:-1] in [
        ((s - 1) / (s ** 2 - 2 * s + 2), -oo),
        ((s - 1) / ((s - 1) ** 2 + 1), -oo),
    ]

    # Fresnel functions
    assert laplace_transform(fresnels(t), t, s) == (
        (
            -sin(s ** 2 / (2 * pi)) * fresnels(s / pi)
            + sin(s ** 2 / (2 * pi)) / 2
            - cos(s ** 2 / (2 * pi)) * fresnelc(s / pi)
            + cos(s ** 2 / (2 * pi)) / 2
        )
        / s,
        0,
        True,
    )
    assert laplace_transform(fresnelc(t), t, s) == (
        (
            sin(s ** 2 / (2 * pi)) * fresnelc(s / pi) / s
            - cos(s ** 2 / (2 * pi)) * fresnels(s / pi) / s
            + sqrt(2) * cos(s ** 2 / (2 * pi) + pi / 4) / (2 * s),
            0,
            True,
        )
    )

    assert LT(Matrix([[exp(t), t * exp(-t)], [t * exp(-t), exp(t)]]), t, s) == Matrix(
        [[(1 / (s - 1), 1, True), ((s + 1) ** (-2), 0, True)], [((s + 1) ** (-2), 0, True), (1 / (s - 1), 1, True)]]
    )

def _construct_symbolic_bu(q, sigma, m):
  return (m - 1) / 2 * sp.erfc(sp.erfcinv(2 * q / (m - 1)) - 1 / sigma)

def test_erfc_evalf():
    assert abs( erfc(Float(2.0)) - 0.00467773 ) < 1E-8 # XXX

def test_erfc_series():
    assert erfc(x).series(x, 0, 7) == 1 - 2*x/sqrt(pi) + \
        2*x**3/3/sqrt(pi) - x**5/5/sqrt(pi) + O(x**7)

def test_meijerint():
    from sympy import symbols, expand, arg

    s, t, mu = symbols("s t mu", real=True)
    assert integrate(
        meijerg([], [], [0], [], s * t) * meijerg([], [], [mu / 2], [-mu / 2], t ** 2 / 4), (t, 0, oo)
    ).is_Piecewise
    s = symbols("s", positive=True)
    assert integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo)) == gamma(s + 1)
    assert integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo), meijerg=True) == gamma(s + 1)
    assert isinstance(integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo), meijerg=False), Integral)

    assert meijerint_indefinite(exp(x), x) == exp(x)

    # TODO what simplifications should be done automatically?
    # This tests "extra case" for antecedents_1.
    a, b = symbols("a b", positive=True)
    assert simplify(meijerint_definite(x ** a, x, 0, b)[0]) == b ** (a + 1) / (a + 1)

    # This tests various conditions and expansions:
    meijerint_definite((x + 1) ** 3 * exp(-x), x, 0, oo) == (16, True)

    # Again, how about simplifications?
    sigma, mu = symbols("sigma mu", positive=True)
    i, c = meijerint_definite(exp(-((x - mu) / (2 * sigma)) ** 2), x, 0, oo)
    assert simplify(i) == sqrt(pi) * sigma * (2 - erfc(mu / (2 * sigma)))
    assert c == True

    i, _ = meijerint_definite(exp(-mu * x) * exp(sigma * x), x, 0, oo)
    # TODO it would be nice to test the condition
    assert simplify(i) == 1 / (mu - sigma)

    # Test substitutions to change limits
    assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
    # Note: causes a NaN in _check_antecedents
    assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
    assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == 1 - exp(-exp(I * arg(x)) * abs(x))

    # Test -oo to oo
    assert meijerint_definite(exp(-x ** 2), x, -oo, oo) == (sqrt(pi), True)
    assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
    assert meijerint_definite(exp(-(2 * x - 3) ** 2), x, -oo, oo) == (sqrt(pi) / 2, True)
    assert meijerint_definite(exp(-abs(2 * x - 3)), x, -oo, oo) == (1, True)
    assert meijerint_definite(exp(-((x - mu) / sigma) ** 2 / 2) / sqrt(2 * pi * sigma ** 2), x, -oo, oo) == (1, True)
    assert meijerint_definite(sinc(x) ** 2, x, -oo, oo) == (pi, True)

    # Test one of the extra conditions for 2 g-functinos
    assert meijerint_definite(exp(-x) * sin(x), x, 0, oo) == (S(1) / 2, True)

    # Test a bug
    def res(n):
        return (1 / (1 + x ** 2)).diff(x, n).subs(x, 1) * (-1) ** n

    for n in range(6):
        assert integrate(exp(-x) * sin(x) * x ** n, (x, 0, oo), meijerg=True) == res(n)

    # This used to test trigexpand... now it is done by linear substitution
    assert simplify(integrate(exp(-x) * sin(x + a), (x, 0, oo), meijerg=True)) == sqrt(2) * sin(a + pi / 4) / 2

    # Test the condition 14 from prudnikov.
    # (This is besselj*besselj in disguise, to stop the product from being
    #  recognised in the tables.)
    a, b, s = symbols("a b s")
    from sympy import And, re

    assert meijerint_definite(
        meijerg([], [], [a / 2], [-a / 2], x / 4) * meijerg([], [], [b / 2], [-b / 2], x / 4) * x ** (s - 1), x, 0, oo
    ) == (
        4
        * 2 ** (2 * s - 2)
        * gamma(-2 * s + 1)
        * gamma(a / 2 + b / 2 + s)
        / (gamma(-a / 2 + b / 2 - s + 1) * gamma(a / 2 - b / 2 - s + 1) * gamma(a / 2 + b / 2 - s + 1)),
        And(0 < -2 * re(4 * s) + 8, 0 < re(a / 2 + b / 2 + s), re(2 * s) < 1),
    )

    # test a bug
    assert integrate(sin(x ** a) * sin(x ** b), (x, 0, oo), meijerg=True) == Integral(
        sin(x ** a) * sin(x ** b), (x, 0, oo)
    )

    # test better hyperexpand
    assert (
        integrate(exp(-x ** 2) * log(x), (x, 0, oo), meijerg=True) == (sqrt(pi) * polygamma(0, S(1) / 2) / 4).expand()
    )

    # Test hyperexpand bug.
    from sympy import lowergamma

    n = symbols("n", integer=True)
    assert simplify(integrate(exp(-x) * x ** n, x, meijerg=True)) == lowergamma(n + 1, x)

    # Test a bug with argument 1/x
    alpha = symbols("alpha", positive=True)
    assert meijerint_definite((2 - x) ** alpha * sin(alpha / x), x, 0, 2) == (
        sqrt(pi)
        * alpha
        * gamma(alpha + 1)
        * meijerg(((), (alpha / 2 + S(1) / 2, alpha / 2 + 1)), ((0, 0, S(1) / 2), (-S(1) / 2,)), alpha ** S(2) / 16)
        / 4,
#.........这里部分代码省略.........