Python sympy.csch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_csch_rewrite():
    x = Symbol("x")
    assert csch(x).rewrite(exp) == 1 / (exp(x) / 2 - exp(-x) / 2) == csch(x).rewrite("tractable")
    assert csch(x).rewrite(cosh) == I / cosh(x + I * pi / 2)
    tanh_half = tanh(S.Half * x)
    assert csch(x).rewrite(tanh) == (1 - tanh_half ** 2) / (2 * tanh_half)
    coth_half = coth(S.Half * x)
    assert csch(x).rewrite(coth) == (coth_half ** 2 - 1) / (2 * coth_half)

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

    assert acsch(-x) == acsch(-x)
    assert acsch(x) == -acsch(-x)

    # values at fixed points
    assert acsch(1) == log(1 + sqrt(2))
    assert acsch(-1) == - log(1 + sqrt(2))
    assert acsch(0) == zoo
    assert acsch(2) == log((1+sqrt(5))/2)
    assert acsch(-2) == - log((1+sqrt(5))/2)

    assert acsch(I) == - I*pi/2
    assert acsch(-I) == I*pi/2
    assert acsch(-I*(sqrt(6) + sqrt(2))) == I*pi / 12
    assert acsch(I*(sqrt(2) + sqrt(6))) == -I*pi / 12
    assert acsch(-I*(1 + sqrt(5))) == I*pi / 10
    assert acsch(I*(1 + sqrt(5))) == -I*pi / 10
    assert acsch(-I*2 / sqrt(2 - sqrt(2))) == I*pi / 8
    assert acsch(I*2 / sqrt(2 - sqrt(2))) == -I*pi / 8
    assert acsch(-I*2) == I*pi / 6
    assert acsch(I*2) == -I*pi / 6
    assert acsch(-I*sqrt(2 + 2/sqrt(5))) == I*pi / 5
    assert acsch(I*sqrt(2 + 2/sqrt(5))) == -I*pi / 5
    assert acsch(-I*sqrt(2)) == I*pi / 4
    assert acsch(I*sqrt(2)) == -I*pi / 4
    assert acsch(-I*(sqrt(5)-1)) == 3*I*pi / 10
    assert acsch(I*(sqrt(5)-1)) == -3*I*pi / 10
    assert acsch(-I*2 / sqrt(3)) == I*pi / 3
    assert acsch(I*2 / sqrt(3)) == -I*pi / 3
    assert acsch(-I*2 / sqrt(2 + sqrt(2))) == 3*I*pi / 8
    assert acsch(I*2 / sqrt(2 + sqrt(2))) == -3*I*pi / 8
    assert acsch(-I*sqrt(2 - 2/sqrt(5))) == 2*I*pi / 5
    assert acsch(I*sqrt(2 - 2/sqrt(5))) == -2*I*pi / 5
    assert acsch(-I*(sqrt(6) - sqrt(2))) == 5*I*pi / 12
    assert acsch(I*(sqrt(6) - sqrt(2))) == -5*I*pi / 12

    # properties
    # acsch(x) == asinh(1/x)
    assert acsch(-I*sqrt(2)) == asinh(I/sqrt(2))
    assert acsch(-I*2 / sqrt(3)) == asinh(I*sqrt(3) / 2)

    # acsch(x) == -I*asin(I/x)
    assert acsch(-I*sqrt(2)) == -I*asin(-1/sqrt(2))
    assert acsch(-I*2 / sqrt(3)) == -I*asin(-sqrt(3)/2)

    # csch(acsch(x)) / x == 1
    assert expand_mul(csch(acsch(-I*(sqrt(6) + sqrt(2)))) / (-I*(sqrt(6) + sqrt(2)))) == 1
    assert expand_mul(csch(acsch(I*(1 + sqrt(5)))) / ((I*(1 + sqrt(5))))) == 1
    assert (csch(acsch(I*sqrt(2 - 2/sqrt(5)))) / (I*sqrt(2 - 2/sqrt(5)))).simplify() == 1
    assert (csch(acsch(-I*sqrt(2 - 2/sqrt(5)))) / (-I*sqrt(2 - 2/sqrt(5)))).simplify() == 1

    # numerical evaluation
    assert str(acsch(5*I+1).n(6)) == '0.0391819 - 0.193363*I'
    assert str(acsch(-5*I+1).n(6)) == '0.0391819 + 0.193363*I'

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_csch_series():
    x = Symbol("x")
    assert csch(x).series(
        x, 0, 10
    ) == 1 / x - x / 6 + 7 * x ** 3 / 360 - 31 * x ** 5 / 15120 + 127 * x ** 7 / 604800 - 73 * x ** 9 / 3421440 + O(
        x ** 10
    )

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_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_csch():
    x, y = symbols('x,y')

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

    assert csch(nan) == nan
    assert csch(zoo) == nan

    assert csch(oo) == 0
    assert csch(-oo) == 0

    assert csch(0) == zoo

    assert csch(-1) == -csch(1)

    assert csch(-x) == -csch(x)
    assert csch(-pi) == -csch(pi)
    assert csch(-2**1024 * E) == -csch(2**1024 * E)

    assert csch(pi*I) == zoo
    assert csch(-pi*I) == zoo
    assert csch(2*pi*I) == zoo
    assert csch(-2*pi*I) == zoo
    assert csch(-3*10**73*pi*I) == zoo
    assert csch(7*10**103*pi*I) == zoo

    assert csch(pi*I/2) == -I
    assert csch(-pi*I/2) == I
    assert csch(5*pi*I/2) == -I
    assert csch(7*pi*I/2) == I

    assert csch(pi*I/3) == -2/sqrt(3)*I
    assert csch(-2*pi*I/3) == 2/sqrt(3)*I

    assert csch(pi*I/4) == -sqrt(2)*I
    assert csch(-pi*I/4) == sqrt(2)*I
    assert csch(7*pi*I/4) == sqrt(2)*I
    assert csch(-3*pi*I/4) == sqrt(2)*I

    assert csch(pi*I/6) == -2*I
    assert csch(-pi*I/6) == 2*I
    assert csch(7*pi*I/6) == 2*I
    assert csch(-7*pi*I/6) == -2*I
    assert csch(-5*pi*I/6) == 2*I

    assert csch(pi*I/105) == -1/sin(pi/105)*I
    assert csch(-pi*I/105) == 1/sin(pi/105)*I

    assert csch(x*I) == -1/sin(x)*I

    assert csch(k*pi*I) == zoo
    assert csch(17*k*pi*I) == zoo

    assert csch(k*pi*I/2) == -1/sin(k*pi/2)*I

    assert csch(n).is_real is True