Python sympy.expand_func函数代码示例

def test_binomial():
    n = Symbol('n', integer=True)
    k = Symbol('k', integer=True)
    u = Symbol('v', negative=True)
    v = Symbol('m', positive=True)

    assert binomial(0, 0) == 1
    assert binomial(1, 1) == 1
    assert binomial(10, 10) == 1
    assert binomial(1, 2) == 0
    assert binomial(1, -1) == 0
    assert binomial(-1, 1) == -1
    assert binomial(-10, 1) == -10
    assert binomial(-10, 7) == -11440
    assert binomial(n, -1) == 0
    assert binomial(n, 0) == 1
    assert expand_func(binomial(n, 1)) == n
    assert expand_func(binomial(n, 2)) == n*(n - 1)/2
    assert expand_func(binomial(n, n - 2)) == n*(n - 1)/2
    assert expand_func(binomial(n, n - 1)) == n
    assert binomial(n, 3).func == binomial
    assert binomial(n, 3).expand(func=True) ==  n**3/6 - n**2/2 + n/3
    assert expand_func(binomial(n, 3)) ==  n*(n - 2)*(n - 1)/6
    assert binomial(n, n) == 1
    assert binomial(n, n + 1) == 0
    assert binomial(n, u) == 0
    assert binomial(n, v).func == binomial
    assert binomial(n, k).func == binomial
    assert binomial(n, n + v) == 0

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))

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

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_beta():
    x, y = Symbol('x'), Symbol('y')

    assert isinstance(beta(x, y), beta)

    assert expand_func(beta(x, y)) == gamma(x)*gamma(y)/gamma(x + y)
    assert expand_func(beta(x, y) - beta(y, x)) == 0  # Symmetric
    assert expand_func(beta(x, y)) == expand_func(beta(x, y + 1) + beta(x + 1, y)).simplify()

    assert diff(beta(x, y), x) == beta(x, y)*(digamma(x) - digamma(x + y))
    assert diff(beta(x, y), y) == beta(x, y)*(digamma(y) - digamma(x + y))

def test_harmonic_rewrite_polygamma():
    n = Symbol("n")
    m = Symbol("m")

    assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma
    assert harmonic(n).rewrite(trigamma) ==  polygamma(0, n + 1) + EulerGamma
    assert harmonic(n).rewrite(polygamma) ==  polygamma(0, n + 1) + EulerGamma

    assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2
    assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)

    assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
    assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n

    assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)

def test_airybiprime():
    z = Symbol('z', real=False)
    t = Symbol('t', negative=True)
    p = Symbol('p', positive=True)

    assert isinstance(airybiprime(z), airybiprime)

    assert airybiprime(0) == 3**(S(1)/6)/gamma(S(1)/3)
    assert airybiprime(oo) == oo
    assert airybiprime(-oo) == 0

    assert diff(airybiprime(z), z) == z*airybi(z)

    assert series(airybiprime(z), z, 0, 3) == (
        3**(S(1)/6)/gamma(S(1)/3) + 3**(S(5)/6)*z**2/(6*gamma(S(2)/3)) + O(z**3))

    assert airybiprime(z).rewrite(hyper) == (
        3**(S(5)/6)*z**2*hyper((), (S(5)/3,), z**S(3)/9)/(6*gamma(S(2)/3)) +
        3**(S(1)/6)*hyper((), (S(1)/3,), z**S(3)/9)/gamma(S(1)/3))

    assert isinstance(airybiprime(z).rewrite(besselj), airybiprime)
    assert airyai(t).rewrite(besselj) == (
        sqrt(-t)*(besselj(-S(1)/3, 2*(-t)**(S(3)/2)/3) +
                  besselj(S(1)/3, 2*(-t)**(S(3)/2)/3))/3)
    assert airybiprime(z).rewrite(besseli) == (
        sqrt(3)*(z**2*besseli(S(2)/3, 2*z**(S(3)/2)/3)/(z**(S(3)/2))**(S(2)/3) +
                 (z**(S(3)/2))**(S(2)/3)*besseli(-S(2)/3, 2*z**(S(3)/2)/3))/3)
    assert airybiprime(p).rewrite(besseli) == (
        sqrt(3)*p*(besseli(-S(2)/3, 2*p**(S(3)/2)/3) + besseli(S(2)/3, 2*p**(S(3)/2)/3))/3)

    assert expand_func(airybiprime(2*(3*z**5)**(S(1)/3))) == (
        sqrt(3)*(z**(S(5)/3)/(z**5)**(S(1)/3) - 1)*airyaiprime(2*3**(S(1)/3)*z**(S(5)/3))/2 +
        (z**(S(5)/3)/(z**5)**(S(1)/3) + 1)*airybiprime(2*3**(S(1)/3)*z**(S(5)/3))/2)

def test_issue1893():
    from sympy import simplify, expand_func, polygamma, gamma

    a = Symbol("a", positive=True)
    assert simplify(expand_func(integrate(exp(-x) * log(x) * x ** a, (x, 0, oo)))) == (a * polygamma(0, a) + 1) * gamma(
        a
    )

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_airyai():
    z = Symbol('z', real=False)
    t = Symbol('t', negative=True)
    p = Symbol('p', positive=True)

    assert isinstance(airyai(z), airyai)

    assert airyai(0) == 3**(S(1)/3)/(3*gamma(S(2)/3))
    assert airyai(oo) == 0
    assert airyai(-oo) == 0

    assert diff(airyai(z), z) == airyaiprime(z)

    assert series(airyai(z), z, 0, 3) == (
        3**(S(5)/6)*gamma(S(1)/3)/(6*pi) - 3**(S(1)/6)*z*gamma(S(2)/3)/(2*pi) + O(z**3))

    assert airyai(z).rewrite(hyper) == (
        -3**(S(2)/3)*z*hyper((), (S(4)/3,), z**S(3)/9)/(3*gamma(S(1)/3)) +
         3**(S(1)/3)*hyper((), (S(2)/3,), z**S(3)/9)/(3*gamma(S(2)/3)))

    assert isinstance(airyai(z).rewrite(besselj), airyai)
    assert airyai(t).rewrite(besselj) == (
        sqrt(-t)*(besselj(-S(1)/3, 2*(-t)**(S(3)/2)/3) +
                  besselj(S(1)/3, 2*(-t)**(S(3)/2)/3))/3)
    assert airyai(z).rewrite(besseli) == (
        -z*besseli(S(1)/3, 2*z**(S(3)/2)/3)/(3*(z**(S(3)/2))**(S(1)/3)) +
         (z**(S(3)/2))**(S(1)/3)*besseli(-S(1)/3, 2*z**(S(3)/2)/3)/3)
    assert airyai(p).rewrite(besseli) == (
        sqrt(p)*(besseli(-S(1)/3, 2*p**(S(3)/2)/3) -
                 besseli(S(1)/3, 2*p**(S(3)/2)/3))/3)

    assert expand_func(airyai(2*(3*z**5)**(S(1)/3))) == (
        -sqrt(3)*(-1 + (z**5)**(S(1)/3)/z**(S(5)/3))*airybi(2*3**(S(1)/3)*z**(S(5)/3))/6 +
         (1 + (z**5)**(S(1)/3)/z**(S(5)/3))*airyai(2*3**(S(1)/3)*z**(S(5)/3))/2)

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 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))

def test_harmonic_rewrite_sum_fail():
    n = Symbol("n")
    m = Symbol("m")

    assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma
    assert harmonic(n).rewrite(trigamma) ==  polygamma(0, n + 1) + EulerGamma
    assert harmonic(n).rewrite(polygamma) ==  polygamma(0, n + 1) + EulerGamma

    assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2
    assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)

    assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
    assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n

    assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)

    _k = Dummy("k")
    assert harmonic(n).rewrite(Sum) == Sum(1/_k, (_k, 1, n))
    assert harmonic(n, m).rewrite(Sum) == Sum(_k**(-m), (_k, 1, n))

def myexpand(func, target):
    expanded = expand_func(func)
    if target is not None:
        return expanded == target
    if expanded == func:  # it didn't expand
        return False

    # check to see that the expanded and original evaluate to the same value
    subs = {}
    for a in func.free_symbols:
        subs[a] = randcplx()
    return abs(func.subs(subs).n() - expanded.replace(exp_polar, exp).subs(subs).n()) < 1e-10

def test_yn():
    z = symbols("z")
    assert myn(0, z) == -cos(z)/z
    assert myn(1, z) == -cos(z)/z**2 - sin(z)/z
    assert myn(2, z) == -((3/z**3 - 1/z)*cos(z) + (3/z**2)*sin(z))
    assert expand_func(yn(n, z)) == yn(n, z)

    # SBFs not defined for complex-valued orders
    assert yn(2+3j, 5.2+0.3j).evalf() == yn(2+3j, 5.2+0.3j)

    assert eq([yn(2, 5.2+0.3j).evalf(10)],
              [0.185250342 + 0.01489557397*I])

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)

def test_jn():
    z = symbols("z")
    assert mjn(0, z) == sin(z)/z
    assert mjn(1, z) == sin(z)/z**2 - cos(z)/z
    assert mjn(2, z) == (3/z**3 - 1/z)*sin(z) - (3/z**2) * cos(z)
    assert mjn(3, z) == (15/z**4 - 6/z**2)*sin(z) + (1/z - 15/z**3)*cos(z)
    assert mjn(4, z) == (1/z + 105/z**5 - 45/z**3)*sin(z) + \
        (-105/z**4 + 10/z**2)*cos(z)
    assert mjn(5, z) == (945/z**6 - 420/z**4 + 15/z**2)*sin(z) + \
        (-1/z - 945/z**5 + 105/z**3)*cos(z)
    assert mjn(6, z) == (-1/z + 10395/z**7 - 4725/z**5 + 210/z**3)*sin(z) + \
        (-10395/z**6 + 1260/z**4 - 21/z**2)*cos(z)

    assert expand_func(jn(n, z)) == jn(n, z)