def test_wallis_product():
    # Wallis product, given in two different forms to ensure that Product
    # can factor simple rational expressions
    A = Product(4*n**2 / (4*n**2 - 1), (n, 1, b))
    B = Product((2*n)*(2*n)/(2*n - 1)/(2*n + 1), (n, 1, b))
    R = pi*gamma(b + 1)**2/(2*gamma(b + S(1)/2)*gamma(b + S(3)/2))
    assert simplify(A.doit()) == R
    assert simplify(B.doit()) == R

def test_wallis_product():
    # Wallis product, given in two different forms to ensure that Product
    # can factor simple rational expressions
    A = Product(4*n**2 / (4*n**2 - 1), (n, 1, b))
    B = Product((2*n)*(2*n)/(2*n - 1)/(2*n + 1), (n, 1, b))
    half = Rational(1, 2)
    R = pi/2 * factorial(b)**2 / factorial(b - half) / factorial(b + half)
    assert simplify(A.doit()) == R
    assert simplify(B.doit()) == R

def test_simple_products():
    assert Product(S.NaN, (x, 1, 3)) is S.NaN
    assert product(S.NaN, (x, 1, 3)) is S.NaN
    assert Product(x, (n, a, a)).doit() == x
    assert Product(x, (x, a, a)).doit() == a
    assert Product(x, (y, 1, a)).doit() == x**a
    lo, hi = 1, 2
    s1 = Product(n, (n, lo, hi))
    s2 = Product(n, (n, hi, lo))
    assert s1 != s2
    assert s1.doit() == s2.doit() == 2
    lo, hi = x, x + 1
    s1 = Product(n, (n, lo, hi))
    s2 = Product(n, (n, hi, lo))
    assert s1 != s2
    assert s1.doit() == s2.doit() == x*(x + 1)
    assert Product(Integral(2*x, (x, 1, y)) + 2*x, (x, 1, 2)).doit() == \
        (y**2 + 1)*(y**2 + 3)
    assert product(2, (n, a, b)) == 2**(b - a + 1)
    assert product(n, (n, 1, b)) == factorial(b)
    assert product(n**3, (n, 1, b)) == factorial(b)**3
    assert product(3**(2 + n), (n, a, b)) \
        == 3**(2*(1 - a + b) + b/2 + (b**2)/2 + a/2 - (a**2)/2)
    assert product(cos(n), (n, 3, 5)) == cos(3)*cos(4)*cos(5)
    assert product(cos(n), (n, x, x + 2)) == cos(x)*cos(x + 1)*cos(x + 2)
    assert isinstance(product(cos(n), (n, x, x + S.Half)), Product)
    # If Product managed to evaluate this one, it most likely got it wrong!
    assert isinstance(Product(n**n, (n, 1, b)), Product)

def test_conjugate_transpose():
    p = Product(x**k, (k, 1, 3))
    assert p.adjoint().doit() == p.doit().adjoint()
    assert p.conjugate().doit() == p.doit().conjugate()
    assert p.transpose().doit() == p.doit().transpose()

    A, B = symbols("A B", commutative=False)
    p = Product(A*B**k, (k, 1, 3))
    assert p.adjoint().doit() == p.doit().adjoint()
    assert p.conjugate().doit() == p.doit().conjugate()
    assert p.transpose().doit() == p.doit().transpose()

def test_issue_9983():
    n = Symbol('n', integer=True, positive=True)
    p = Product(1 + 1/n**(S(2)/3), (n, 1, oo))
    assert p.is_convergent() is S.false
    assert product(1 + 1/n**(S(2)/3), (n, 1, oo)) == p.doit()

def test_issue_13546():
    n = Symbol('n')
    k = Symbol('k')
    p = Product(n + 1 / 2**k, (k, 0, n-1)).doit()
    assert p.subs(n, 2).doit() == S(15)/2