Python sympy.sec函数代码示例

def test_hyper_as_trig():
    from sympy.simplify.fu import _osborne as o, _osbornei as i, TR12

    eq = sinh(x)**2 + cosh(x)**2
    t, f = hyper_as_trig(eq)
    assert f(fu(t)) == cosh(2*x)
    e, f = hyper_as_trig(tanh(x + y))
    assert f(TR12(e)) == (tanh(x) + tanh(y))/(tanh(x)*tanh(y) + 1)

    d = Dummy()
    assert o(sinh(x), d) == I*sin(x*d)
    assert o(tanh(x), d) == I*tan(x*d)
    assert o(coth(x), d) == cot(x*d)/I
    assert o(cosh(x), d) == cos(x*d)
    assert o(sech(x), d) == sec(x*d)
    assert o(csch(x), d) == csc(x*d)/I
    for func in (sinh, cosh, tanh, coth, sech, csch):
        h = func(pi)
        assert i(o(h, d), d) == h
    # /!\ the _osborne functions are not meant to work
    # in the o(i(trig, d), d) direction so we just check
    # that they work as they are supposed to work
    assert i(cos(x*y + z), y) == cosh(x + z*I)
    assert i(sin(x*y + z), y) == sinh(x + z*I)/I
    assert i(tan(x*y + z), y) == tanh(x + z*I)/I
    assert i(cot(x*y + z), y) == coth(x + z*I)*I
    assert i(sec(x*y + z), y) == sech(x + z*I)
    assert i(csc(x*y + z), y) == csch(x + z*I)*I

def trig_substitution_rule(integral):
    integrand, symbol = integral
    A = sympy.Wild('a', exclude=[0, symbol])
    B = sympy.Wild('b', exclude=[0, symbol])
    theta = sympy.Dummy("theta")
    target_pattern = A + B*symbol**2

    matches = integrand.find(target_pattern)
    for expr in matches:
        match = expr.match(target_pattern)
        a = match.get(A, ZERO)
        b = match.get(B, ZERO)

        a_positive = ((a.is_number and a > 0) or a.is_positive)
        b_positive = ((b.is_number and b > 0) or b.is_positive)
        a_negative = ((a.is_number and a < 0) or a.is_negative)
        b_negative = ((b.is_number and b < 0) or b.is_negative)
        x_func = None
        if a_positive and b_positive:
            # a**2 + b*x**2. Assume sec(theta) > 0, -pi/2 < theta < pi/2
            x_func = (sympy.sqrt(a)/sympy.sqrt(b)) * sympy.tan(theta)
            # Do not restrict the domain: tan(theta) takes on any real
            # value on the interval -pi/2 < theta < pi/2 so x takes on
            # any value
            restriction = True
        elif a_positive and b_negative:
            # a**2 - b*x**2. Assume cos(theta) > 0, -pi/2 < theta < pi/2
            constant = sympy.sqrt(a)/sympy.sqrt(-b)
            x_func = constant * sympy.sin(theta)
            restriction = sympy.And(symbol > -constant, symbol < constant)
        elif a_negative and b_positive:
            # b*x**2 - a**2. Assume sin(theta) > 0, 0 < theta < pi
            constant = sympy.sqrt(-a)/sympy.sqrt(b)
            x_func = constant * sympy.sec(theta)
            restriction = sympy.And(symbol > -constant, symbol < constant)
        if x_func:
            # Manually simplify sqrt(trig(theta)**2) to trig(theta)
            # Valid due to assumed domain restriction
            substitutions = {}
            for f in [sympy.sin, sympy.cos, sympy.tan,
                      sympy.sec, sympy.csc, sympy.cot]:
                substitutions[sympy.sqrt(f(theta)**2)] = f(theta)
                substitutions[sympy.sqrt(f(theta)**(-2))] = 1/f(theta)

            replaced = integrand.subs(symbol, x_func).trigsimp()
            replaced = replaced.subs(substitutions)
            if not replaced.has(symbol):
                replaced *= manual_diff(x_func, theta)
                replaced = replaced.trigsimp()
                secants = replaced.find(1/sympy.cos(theta))
                if secants:
                    replaced = replaced.xreplace({
                        1/sympy.cos(theta): sympy.sec(theta)
                    })

                substep = integral_steps(replaced, theta)
                if not contains_dont_know(substep):
                    return TrigSubstitutionRule(
                        theta, x_func, replaced, substep, restriction,
                        integrand, symbol)

def test_periodicity():
    x = Symbol('x')
    y = Symbol('y')

    assert periodicity(sin(2*x), x) == pi
    assert periodicity((-2)*tan(4*x), x) == pi/4
    assert periodicity(sin(x)**2, x) == 2*pi
    assert periodicity(3**tan(3*x), x) == pi/3
    assert periodicity(tan(x)*cos(x), x) == 2*pi
    assert periodicity(sin(x)**(tan(x)), x) == 2*pi
    assert periodicity(tan(x)*sec(x), x) == 2*pi
    assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2
    assert periodicity(tan(x) + cot(x), x) == pi
    assert periodicity(sin(x) - cos(2*x), x) == 2*pi
    assert periodicity(sin(x) - 1, x) == 2*pi
    assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi
    assert periodicity(exp(sin(x)), x) == 2*pi
    assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi
    assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi
    assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi
    assert periodicity(tan(sin(2*x)), x) == pi
    assert periodicity(2*tan(x)**2, x) == pi

    assert periodicity(sin(x)**2 + cos(x)**2, x) == S.Zero
    assert periodicity(tan(x), y) == S.Zero

    assert periodicity(exp(x), x) is None
    assert periodicity(log(x), x) is None
    assert periodicity(exp(x)**sin(x), x) is None
    assert periodicity(sin(x)**y, y) is None

def trig_rule(integral):
    integrand, symbol = integral
    if isinstance(integrand, sympy.sin) or isinstance(integrand, sympy.cos):
        arg = integrand.args[0]

        if not isinstance(arg, sympy.Symbol):
            return  # perhaps a substitution can deal with it

        if isinstance(integrand, sympy.sin):
            func = "sin"
        else:
            func = "cos"

        return TrigRule(func, arg, integrand, symbol)

    if integrand == sympy.sec(symbol) ** 2:
        return TrigRule("sec**2", symbol, integrand, symbol)
    elif integrand == sympy.csc(symbol) ** 2:
        return TrigRule("csc**2", symbol, integrand, symbol)

    if isinstance(integrand, sympy.tan):
        rewritten = sympy.sin(*integrand.args) / sympy.cos(*integrand.args)
    elif isinstance(integrand, sympy.cot):
        rewritten = sympy.cos(*integrand.args) / sympy.sin(*integrand.args)
    elif isinstance(integrand, sympy.sec):
        arg = integrand.args[0]
        rewritten = (sympy.sec(arg) ** 2 + sympy.tan(arg) * sympy.sec(arg)) / (sympy.sec(arg) + sympy.tan(arg))
    elif isinstance(integrand, sympy.csc):
        arg = integrand.args[0]
        rewritten = (sympy.csc(arg) ** 2 + sympy.cot(arg) * sympy.csc(arg)) / (sympy.csc(arg) + sympy.cot(arg))
    else:
        return

    return RewriteRule(rewritten, integral_steps(rewritten, symbol), integrand, symbol)

def test_periodicity_check():
    x = Symbol('x')
    y = Symbol('y')

    assert periodicity(tan(x), x, check=True) == pi
    assert periodicity(sin(x) + cos(x), x, check=True) == 2*pi
    assert periodicity(sec(x), x) == 2*pi
    assert periodicity(sin(x*y), x) == 2*pi/abs(y)
    assert periodicity(Abs(sec(sec(x))), x) == pi

def test_manualintegrate_trigonometry():
    assert manualintegrate(sin(x), x) == -cos(x)
    assert manualintegrate(tan(x), x) == -log(cos(x))

    assert manualintegrate(sec(x), x) == log(sec(x) + tan(x))
    assert manualintegrate(csc(x), x) == -log(csc(x) + cot(x))

    assert manualintegrate(sin(x) * cos(x), x) in [sin(x) ** 2 / 2, -cos(x)**2 / 2]
    assert manualintegrate(-sec(x) * tan(x), x) == -sec(x)
    assert manualintegrate(csc(x) * cot(x), x) == -csc(x)

def test_trigintegrate_mixed():
    assert trigintegrate(sin(x)*sec(x), x) == -log(sin(x)**2 - 1)/2
    assert trigintegrate(sin(x)*csc(x), x) == x
    assert trigintegrate(sin(x)*cot(x), x) == sin(x)

    assert trigintegrate(cos(x)*sec(x), x) == x
    assert trigintegrate(cos(x)*csc(x), x) == log(cos(x)**2 - 1)/2
    assert trigintegrate(cos(x)*tan(x), x) == -cos(x)
    assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \
        - log(cos(x) + 1)/2 + cos(x)

def test_sech():
    x, y = symbols('x, y')

    k = Symbol('k', integer=True)
    n = Symbol('n', positive=True)

    assert sech(nan) == nan
    assert sech(zoo) == nan

    assert sech(oo) == 0
    assert sech(-oo) == 0

    assert sech(0) == 1

    assert sech(-1) == sech(1)
    assert sech(-x) == sech(x)

    assert sech(pi*I) == sec(pi)

    assert sech(-pi*I) == sec(pi)
    assert sech(-2**1024 * E) == sech(2**1024 * E)

    assert sech(pi*I/2) == zoo
    assert sech(-pi*I/2) == zoo
    assert sech((-3*10**73 + 1)*pi*I/2) == zoo
    assert sech((7*10**103 + 1)*pi*I/2) == zoo

    assert sech(pi*I) == -1
    assert sech(-pi*I) == -1
    assert sech(5*pi*I) == -1
    assert sech(8*pi*I) == 1

    assert sech(pi*I/3) == 2
    assert sech(-2*pi*I/3) == -2

    assert sech(pi*I/4) == sqrt(2)
    assert sech(-pi*I/4) == sqrt(2)
    assert sech(5*pi*I/4) == -sqrt(2)
    assert sech(-5*pi*I/4) == -sqrt(2)

    assert sech(pi*I/6) == 2/sqrt(3)
    assert sech(-pi*I/6) == 2/sqrt(3)
    assert sech(7*pi*I/6) == -2/sqrt(3)
    assert sech(-5*pi*I/6) == -2/sqrt(3)

    assert sech(pi*I/105) == 1/cos(pi/105)
    assert sech(-pi*I/105) == 1/cos(pi/105)

    assert sech(x*I) == 1/cos(x)

    assert sech(k*pi*I) == 1/cos(k*pi)
    assert sech(17*k*pi*I) == 1/cos(17*k*pi)

    assert sech(n).is_real is True

def test_periodicity():
    x = Symbol('x')
    y = Symbol('y')

    assert periodicity(sin(2*x), x) == pi
    assert periodicity((-2)*tan(4*x), x) == pi/4
    assert periodicity(sin(x)**2, x) == 2*pi
    assert periodicity(3**tan(3*x), x) == pi/3
    assert periodicity(tan(x)*cos(x), x) == 2*pi
    assert periodicity(sin(x)**(tan(x)), x) == 2*pi
    assert periodicity(tan(x)*sec(x), x) == 2*pi
    assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2
    assert periodicity(tan(x) + cot(x), x) == pi
    assert periodicity(sin(x) - cos(2*x), x) == 2*pi
    assert periodicity(sin(x) - 1, x) == 2*pi
    assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi
    assert periodicity(exp(sin(x)), x) == 2*pi
    assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi
    assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi
    assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi
    assert periodicity(tan(sin(2*x)), x) == pi
    assert periodicity(2*tan(x)**2, x) == pi
    assert periodicity(sin(x%4), x) == 4
    assert periodicity(sin(x)%4, x) == 2*pi
    assert periodicity(tan((3*x-2)%4), x) == 4/3
    assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1)
    assert periodicity((x**2+1) % x, x) == None

    assert periodicity(sin(x)**2 + cos(x)**2, x) == S.Zero
    assert periodicity(tan(x), y) == S.Zero

    assert periodicity(exp(x), x) is None
    assert periodicity(log(x), x) is None
    assert periodicity(exp(x)**sin(x), x) is None
    assert periodicity(sin(x)**y, y) is None

    assert periodicity(Abs(sin(Abs(sin(x)))),x) == pi
    assert all(periodicity(Abs(f(x)),x) == pi for f in (
        cos, sin, sec, csc, tan, cot))
    assert periodicity(Abs(sin(tan(x))), x) == pi
    assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2*pi
    assert periodicity(sin(x) > S.Half, x) is 2*pi

    assert periodicity(x > 2, x) is None
    assert periodicity(x**3 - x**2 + 1, x) is None
    assert periodicity(Abs(x), x) is None
    assert periodicity(Abs(x**2 - 1), x) is None

    assert periodicity((x**2 + 4)%2, x) is None
    assert periodicity((E**x)%3, x) is None

def test_manualintegrate_trigpowers():
    assert manualintegrate(sin(x)**2 * cos(x), x) == sin(x)**3 / 3
    assert manualintegrate(sin(x)**2 * cos(x) **2, x) == \
        x / 8 - sin(4*x) / 32
    assert manualintegrate(sin(x) * cos(x)**3, x) == -cos(x)**4 / 4
    assert manualintegrate(sin(x)**3 * cos(x)**2, x) == \
        cos(x)**5 / 5 - cos(x)**3 / 3

    assert manualintegrate(tan(x)**3 * sec(x), x) == sec(x)**3/3 - sec(x)
    assert manualintegrate(tan(x) * sec(x) **2, x) == sec(x)**2/2

    assert manualintegrate(cot(x)**5 * csc(x), x) == \
        -csc(x)**5/5 + 2*csc(x)**3/3 - csc(x)
    assert manualintegrate(cot(x)**2 * csc(x)**6, x) == \
        -cot(x)**7/7 - 2*cot(x)**5/5 - cot(x)**3/3

def test_conv7():
    x = Symbol("x")
    y = Symbol("y")
    assert sin(x/3) == sin(sympy.Symbol("x") / 3)
    assert cos(x/3) == cos(sympy.Symbol("x") / 3)
    assert tan(x/3) == tan(sympy.Symbol("x") / 3)
    assert cot(x/3) == cot(sympy.Symbol("x") / 3)
    assert csc(x/3) == csc(sympy.Symbol("x") / 3)
    assert sec(x/3) == sec(sympy.Symbol("x") / 3)
    assert asin(x/3) == asin(sympy.Symbol("x") / 3)
    assert acos(x/3) == acos(sympy.Symbol("x") / 3)
    assert atan(x/3) == atan(sympy.Symbol("x") / 3)
    assert acot(x/3) == acot(sympy.Symbol("x") / 3)
    assert acsc(x/3) == acsc(sympy.Symbol("x") / 3)
    assert asec(x/3) == asec(sympy.Symbol("x") / 3)

    assert sin(x/3)._sympy_() == sympy.sin(sympy.Symbol("x") / 3)
    assert sin(x/3)._sympy_() != sympy.cos(sympy.Symbol("x") / 3)
    assert cos(x/3)._sympy_() == sympy.cos(sympy.Symbol("x") / 3)
    assert tan(x/3)._sympy_() == sympy.tan(sympy.Symbol("x") / 3)
    assert cot(x/3)._sympy_() == sympy.cot(sympy.Symbol("x") / 3)
    assert csc(x/3)._sympy_() == sympy.csc(sympy.Symbol("x") / 3)
    assert sec(x/3)._sympy_() == sympy.sec(sympy.Symbol("x") / 3)
    assert asin(x/3)._sympy_() == sympy.asin(sympy.Symbol("x") / 3)
    assert acos(x/3)._sympy_() == sympy.acos(sympy.Symbol("x") / 3)
    assert atan(x/3)._sympy_() == sympy.atan(sympy.Symbol("x") / 3)
    assert acot(x/3)._sympy_() == sympy.acot(sympy.Symbol("x") / 3)
    assert acsc(x/3)._sympy_() == sympy.acsc(sympy.Symbol("x") / 3)
    assert asec(x/3)._sympy_() == sympy.asec(sympy.Symbol("x") / 3)

def test_stationary_points():
    x, y = symbols('x y')

    assert stationary_points(sin(x), x, Interval(-pi/2, pi/2)
        ) == {-pi/2, pi/2}
    assert  stationary_points(sin(x), x, Interval.Ropen(0, pi/4)
        ) == EmptySet()
    assert stationary_points(tan(x), x,
        ) == EmptySet()
    assert stationary_points(sin(x)*cos(x), x, Interval(0, pi)
        ) == {pi/4, 3*pi/4}
    assert stationary_points(sec(x), x, Interval(0, pi)
        ) == {0, pi}
    assert stationary_points((x+3)*(x-2), x
        ) == FiniteSet(-S.Half)
    assert stationary_points((x + 3)/(x - 2), x, Interval(-5, 5)
        ) == EmptySet()
    assert stationary_points((x**2+3)/(x-2), x
        ) == {2 - sqrt(7), 2 + sqrt(7)}
    assert stationary_points((x**2+3)/(x-2), x, Interval(0, 5)
        ) == {2 + sqrt(7)}
    assert stationary_points(x**4 + x**3 - 5*x**2, x, S.Reals
        ) == FiniteSet(-2, 0, S(5)/4)
    assert stationary_points(exp(x), x
        ) == EmptySet()
    assert stationary_points(log(x) - x, x, S.Reals
        ) == {1}
    assert stationary_points(cos(x), x, Union(Interval(0, 5), Interval(-6, -3))
        ) == {0, -pi, pi}
    assert stationary_points(y, x, S.Reals
        ) == S.Reals

def eval_trigsubstitution(theta, func, rewritten, substep, restriction, integrand, symbol):
    func = func.subs(sympy.sec(theta), 1/sympy.cos(theta))

    trig_function = list(func.find(TrigonometricFunction))
    assert len(trig_function) == 1
    trig_function = trig_function[0]
    relation = sympy.solve(symbol - func, trig_function)
    assert len(relation) == 1
    numer, denom = sympy.fraction(relation[0])

    if isinstance(trig_function, sympy.sin):
        opposite = numer
        hypotenuse = denom
        adjacent = sympy.sqrt(denom**2 - numer**2)
        inverse = sympy.asin(relation[0])
    elif isinstance(trig_function, sympy.cos):
        adjacent = numer
        hypotenuse = denom
        opposite = sympy.sqrt(denom**2 - numer**2)
        inverse = sympy.acos(relation[0])
    elif isinstance(trig_function, sympy.tan):
        opposite = numer
        adjacent = denom
        hypotenuse = sympy.sqrt(denom**2 + numer**2)
        inverse = sympy.atan(relation[0])

    substitution = [
        (sympy.sin(theta), opposite/hypotenuse),
        (sympy.cos(theta), adjacent/hypotenuse),
        (sympy.tan(theta), opposite/adjacent),
        (theta, inverse)
    ]
    return sympy.Piecewise(
        (_manualintegrate(substep).subs(substitution).trigsimp(), restriction)
    )

def test_periodicity_check():
    x = Symbol('x')
    y = Symbol('y')

    assert periodicity(tan(x), x, check=True) == pi
    assert periodicity(sin(x) + cos(x), x, check=True) == 2*pi
    raises(NotImplementedError, lambda: periodicity(sec(x), x, check=True))
    raises(NotImplementedError, lambda: periodicity(sin(x*y), x, check=True))

def eval_trig(func, arg, integrand, symbol):
    if func == 'sin':
        return -sympy.cos(arg)
    elif func == 'cos':
        return sympy.sin(arg)
    elif func == 'sec*tan':
        return sympy.sec(arg)
    elif func == 'csc*cot':
        return sympy.csc(arg)