Python sympy.sech函数代码示例

def test_simplifications():
    x = Symbol('x')
    assert sinh(asinh(x)) == x
    assert sinh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1)
    assert sinh(atanh(x)) == x/sqrt(1 - x**2)
    assert sinh(acoth(x)) == 1/(sqrt(x - 1) * sqrt(x + 1))

    assert cosh(asinh(x)) == sqrt(1 + x**2)
    assert cosh(acosh(x)) == x
    assert cosh(atanh(x)) == 1/sqrt(1 - x**2)
    assert cosh(acoth(x)) == x/(sqrt(x - 1) * sqrt(x + 1))

    assert tanh(asinh(x)) == x/sqrt(1 + x**2)
    assert tanh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1) / x
    assert tanh(atanh(x)) == x
    assert tanh(acoth(x)) == 1/x

    assert coth(asinh(x)) == sqrt(1 + x**2)/x
    assert coth(acosh(x)) == x/(sqrt(x - 1) * sqrt(x + 1))
    assert coth(atanh(x)) == 1/x
    assert coth(acoth(x)) == x

    assert csch(asinh(x)) == 1/x
    assert csch(acosh(x)) == 1/(sqrt(x - 1) * sqrt(x + 1))
    assert csch(atanh(x)) == sqrt(1 - x**2)/x
    assert csch(acoth(x)) == sqrt(x - 1) * sqrt(x + 1)

    assert sech(asinh(x)) == 1/sqrt(1 + x**2)
    assert sech(acosh(x)) == 1/x
    assert sech(atanh(x)) == sqrt(1 - x**2)
    assert sech(acoth(x)) == sqrt(x - 1) * sqrt(x + 1)/x

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 test_sech_rewrite():
    x = Symbol("x")
    assert sech(x).rewrite(exp) == 1 / (exp(x) / 2 + exp(-x) / 2) == sech(x).rewrite("tractable")
    assert sech(x).rewrite(sinh) == I / sinh(x + I * pi / 2)
    tanh_half = tanh(S.Half * x) ** 2
    assert sech(x).rewrite(tanh) == (1 - tanh_half) / (1 + tanh_half)
    coth_half = coth(S.Half * x) ** 2
    assert sech(x).rewrite(coth) == (coth_half - 1) / (coth_half + 1)

def test_derivs():
    x = Symbol('x')
    assert coth(x).diff(x) == -sinh(x)**(-2)
    assert sinh(x).diff(x) == cosh(x)
    assert cosh(x).diff(x) == sinh(x)
    assert tanh(x).diff(x) == -tanh(x)**2 + 1
    assert csch(x).diff(x) == -coth(x)*csch(x)
    assert sech(x).diff(x) == -tanh(x)*sech(x)
    assert acoth(x).diff(x) == 1/(-x**2 + 1)
    assert asinh(x).diff(x) == 1/sqrt(x**2 + 1)
    assert acosh(x).diff(x) == 1/sqrt(x**2 - 1)
    assert atanh(x).diff(x) == 1/(-x**2 + 1)

def test_sign_assumptions():
    p = Symbol('p', positive=True)
    n = Symbol('n', negative=True)
    assert sinh(n).is_negative is True
    assert sinh(p).is_positive is True
    assert cosh(n).is_positive is True
    assert cosh(p).is_positive is True
    assert tanh(n).is_negative is True
    assert tanh(p).is_positive is True
    assert csch(n).is_negative is True
    assert csch(p).is_positive is True
    assert sech(n).is_positive is True
    assert sech(p).is_positive is True
    assert coth(n).is_negative is True
    assert coth(p).is_positive is True

def test_rewrite_trigh():
    # if this import passes then the test below should also pass
    from sympy import sech
    assert solveset_real(sinh(x) + sech(x), x) == FiniteSet(
        2*atanh(-S.Half + sqrt(5)/2 - sqrt(-2*sqrt(5) + 2)/2),
        2*atanh(-S.Half + sqrt(5)/2 + sqrt(-2*sqrt(5) + 2)/2),
        2*atanh(-sqrt(5)/2 - S.Half + sqrt(2 + 2*sqrt(5))/2),
        2*atanh(-sqrt(2 + 2*sqrt(5))/2 - sqrt(5)/2 - S.Half))

def test_real_assumptions():
    z = Symbol('z', real=False)
    assert sinh(z).is_real is None
    assert cosh(z).is_real is None
    assert tanh(z).is_real is None
    assert sech(z).is_real is None
    assert csch(z).is_real is None
    assert coth(z).is_real is None

def test_asech():
    x = Symbol('x')

    assert asech(-x) == asech(-x)

    # values at fixed points
    assert asech(1) == 0
    assert asech(-1) == pi*I
    assert asech(0) == oo
    assert asech(2) == I*pi/3
    assert asech(-2) == 2*I*pi / 3

    # at infinites
    assert asech(oo) == I*pi/2
    assert asech(-oo) == I*pi/2
    assert asech(zoo) == nan

    assert asech(I) == log(1 + sqrt(2)) - I*pi/2
    assert asech(-I) == log(1 + sqrt(2)) + I*pi/2
    assert asech(sqrt(2) - sqrt(6)) == 11*I*pi / 12
    assert asech(sqrt(2 - 2/sqrt(5))) == I*pi / 10
    assert asech(-sqrt(2 - 2/sqrt(5))) == 9*I*pi / 10
    assert asech(2 / sqrt(2 + sqrt(2))) == I*pi / 8
    assert asech(-2 / sqrt(2 + sqrt(2))) == 7*I*pi / 8
    assert asech(sqrt(5) - 1) == I*pi / 5
    assert asech(1 - sqrt(5)) == 4*I*pi / 5
    assert asech(-sqrt(2*(2 + sqrt(2)))) == 5*I*pi / 8

    # properties
    # asech(x) == acosh(1/x)
    assert asech(sqrt(2)) == acosh(1/sqrt(2))
    assert asech(2/sqrt(3)) == acosh(sqrt(3)/2)
    assert asech(2/sqrt(2 + sqrt(2))) == acosh(sqrt(2 + sqrt(2))/2)
    assert asech(S(2)) == acosh(1/S(2))

    # asech(x) == I*acos(1/x)
    # (Note: the exact formula is asech(x) == +/- I*acos(1/x))
    assert asech(-sqrt(2)) == I*acos(-1/sqrt(2))
    assert asech(-2/sqrt(3)) == I*acos(-sqrt(3)/2)
    assert asech(-S(2)) == I*acos(-S.Half)
    assert asech(-2/sqrt(2)) == I*acos(-sqrt(2)/2)

    # sech(asech(x)) / x == 1
    assert expand_mul(sech(asech(sqrt(6) - sqrt(2))) / (sqrt(6) - sqrt(2))) == 1
    assert expand_mul(sech(asech(sqrt(6) + sqrt(2))) / (sqrt(6) + sqrt(2))) == 1
    assert (sech(asech(sqrt(2 + 2/sqrt(5)))) / (sqrt(2 + 2/sqrt(5)))).simplify() == 1
    assert (sech(asech(-sqrt(2 + 2/sqrt(5)))) / (-sqrt(2 + 2/sqrt(5)))).simplify() == 1
    assert (sech(asech(sqrt(2*(2 + sqrt(2))))) / (sqrt(2*(2 + sqrt(2))))).simplify() == 1
    assert expand_mul(sech(asech((1 + sqrt(5)))) / ((1 + sqrt(5)))) == 1
    assert expand_mul(sech(asech((-1 - sqrt(5)))) / ((-1 - sqrt(5)))) == 1
    assert expand_mul(sech(asech((-sqrt(6) - sqrt(2)))) / ((-sqrt(6) - sqrt(2)))) == 1

    # numerical evaluation
    assert str(asech(5*I).n(6)) == '0.19869 - 1.5708*I'
    assert str(asech(-5*I).n(6)) == '0.19869 + 1.5708*I'

def test_complex():
    a, b = symbols('a,b', real=True)
    z = a + b*I
    for func in [sinh, cosh, tanh, coth, sech, csch]:
        assert func(z).conjugate() == func(a - b*I)
    for deep in [True, False]:
        assert sinh(z).expand(
            complex=True, deep=deep) == sinh(a)*cos(b) + I*cosh(a)*sin(b)
        assert cosh(z).expand(
            complex=True, deep=deep) == cosh(a)*cos(b) + I*sinh(a)*sin(b)
        assert tanh(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
            a)/(cos(b)**2 + sinh(a)**2) + I*sin(b)*cos(b)/(cos(b)**2 + sinh(a)**2)
        assert coth(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
            a)/(sin(b)**2 + sinh(a)**2) - I*sin(b)*cos(b)/(sin(b)**2 + sinh(a)**2)
        assert csch(z).expand(complex=True, deep=deep) == cos(b) * sinh(a) / (sin(b)**2\
            *cosh(a)**2 + cos(b)**2 * sinh(a)**2) - I*sin(b) * cosh(a) / (sin(b)**2\
            *cosh(a)**2 + cos(b)**2 * sinh(a)**2)
        assert sech(z).expand(complex=True, deep=deep) == cos(b) * cosh(a) / (sin(b)**2\
            *sinh(a)**2 + cos(b)**2 * cosh(a)**2) - I*sin(b) * sinh(a) / (sin(b)**2\
            *sinh(a)**2 + cos(b)**2 * cosh(a)**2)

def test_sech_series():
    x = Symbol('x')
    assert sech(x).series(x, 0, 10) == \
        1 - x**2/2 + 5*x**4/24 - 61*x**6/720 + 277*x**8/8064 + O(x**10)

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