Python sympy.tanh函数代码示例

def test_tan_rewrite():
    neg_exp, pos_exp = exp(-x*I), exp(x*I)
    assert tan(x).rewrite(exp) == I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
    assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x)
    assert tan(x).rewrite(cos) == -cos(x + S.Pi/2)/cos(x)
    assert tan(x).rewrite(cot) == 1/cot(x)
    assert tan(sinh(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n()
    assert tan(cosh(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n()
    assert tan(tanh(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n()
    assert tan(coth(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n()
    assert tan(sin(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n()
    assert tan(cos(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n()
    assert tan(tan(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n()
    assert tan(cot(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n()
    assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
    assert 0 == (cos(pi/15)*tan(pi/15) - sin(pi/15)).rewrite(pow)
    assert tan(pi/19).rewrite(pow) == tan(pi/19)
    assert tan(8*pi/19).rewrite(sqrt) == tan(8*pi/19)

def test_evalc():
    x = Symbol("x", real=True)
    y = Symbol("y", real=True)
    z = Symbol("z")
    assert ((x+I*y)**2).expand(complex=True) == x**2+2*I*x*y - y**2
    assert expand_complex(z**(2*I)) == I*im(z**(2*I)) + re(z**(2*I))

    assert exp(I*x) != cos(x)+I*sin(x)
    assert exp(I*x).expand(complex=True) == cos(x)+I*sin(x)
    assert exp(I*x+y).expand(complex=True) == exp(y)*cos(x)+I*sin(x)*exp(y)

    assert sin(I*x).expand(complex=True) == I * sinh(x)
    assert sin(x+I*y).expand(complex=True) == sin(x)*cosh(y) + \
            I * sinh(y) * cos(x)

    assert cos(I*x).expand(complex=True) == cosh(x)
    assert cos(x+I*y).expand(complex=True) == cos(x)*cosh(y) - \
            I * sinh(y) * sin(x)

    assert tan(I*x).expand(complex=True) == tanh(x) * I
    assert tan(x+I*y).expand(complex=True) == \
            ((sin(x)*cos(x) + I*cosh(y)*sinh(y)) / (cos(x)**2 + sinh(y)**2)).expand()

    assert sinh(I*x).expand(complex=True) == I * sin(x)
    assert sinh(x+I*y).expand(complex=True) == sinh(x)*cos(y) + \
            I * sin(y) * cosh(x)

    assert cosh(I*x).expand(complex=True) == cos(x)
    assert cosh(x+I*y).expand(complex=True) == cosh(x)*cos(y) + \
            I * sin(y) * sinh(x)

    assert tanh(I*x).expand(complex=True) == tan(x) * I
    assert tanh(x+I*y).expand(complex=True) == \
            ((sinh(x)*cosh(x) + I*cos(y)*sin(y)) / (sinh(x)**2 + cos(y)**2)).expand()

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_tanh_rewrite():
    x = Symbol('x')
    assert tanh(x).rewrite(exp) == (exp(x) - exp(-x))/(exp(x) + exp(-x)) \
        == tanh(x).rewrite('tractable')
    assert tanh(x).rewrite(sinh) == I*sinh(x)/sinh(I*pi/2 - x)
    assert tanh(x).rewrite(cosh) == I*cosh(I*pi/2 - x)/cosh(x)
    assert tanh(x).rewrite(coth) == 1/coth(x)

def test_function_series3():
    """
    Test our easy "tanh" function.

    This test tests two things:
      * that the Function interface works as expected and it's easy to use
      * that the general algorithm for the series expansion works even when the
        derivative is defined recursively in terms of the original function,
        since tanh(x).diff(x) == 1-tanh(x)**2
    """

    class mytanh(Function):
        nargs = 1

        def fdiff(self, argindex = 1):
            return 1-mytanh(self.args[0])**2

        @classmethod
        def canonize(cls, arg):
            arg = sympify(arg)
            if arg == 0:
                return sympify(0)

    e = tanh(x)
    f = mytanh(x)
    assert tanh(x).series(x, 0, 6) == mytanh(x).series(x, 0, 6)

def test_exp_rewrite():
    assert exp(x).rewrite(sin) == sinh(x) + cosh(x)
    assert exp(x*I).rewrite(cos) == cos(x) + I*sin(x)
    assert exp(1).rewrite(cos) == sinh(1) + cosh(1)
    assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
    assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
    assert exp(x).rewrite(tanh) == (1 + tanh(x/2))/(1 - tanh(x/2))

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

    r = Symbol('r', real=True)

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

    assert tan(nan) == nan

    assert tan(oo*I) == I
    assert tan(-oo*I) == -I

    assert tan(0) == 0

    assert tan(atan(x)) == x
    assert tan(asin(x)) == x / sqrt(1 - x**2)
    assert tan(acos(x)) == sqrt(1 - x**2) / x
    assert tan(acot(x)) == 1 / x
    assert tan(atan2(y, x)) == y/x

    assert tan(pi*I) == tanh(pi)*I
    assert tan(-pi*I) == -tanh(pi)*I
    assert tan(-2*I) == -tanh(2)*I

    assert tan(pi) == 0
    assert tan(-pi) == 0
    assert tan(2*pi) == 0
    assert tan(-2*pi) == 0
    assert tan(-3*10**73*pi) == 0

    assert tan(pi/2) == zoo
    assert tan(3*pi/2) == zoo

    assert tan(pi/3) == sqrt(3)
    assert tan(-2*pi/3) == sqrt(3)

    assert tan(pi/4) == S.One
    assert tan(-pi/4) == -S.One
    assert tan(17*pi/4) == S.One
    assert tan(-3*pi/4) == S.One

    assert tan(pi/6) == 1/sqrt(3)
    assert tan(-pi/6) == -1/sqrt(3)
    assert tan(7*pi/6) == 1/sqrt(3)
    assert tan(-5*pi/6) == 1/sqrt(3)

    assert tan(x*I) == tanh(x)*I

    assert tan(k*pi) == 0
    assert tan(17*k*pi) == 0

    assert tan(k*pi*I) == tanh(k*pi)*I

    assert tan(r).is_real is True

    assert tan(10*pi/7) == tan(3*pi/7)
    assert tan(11*pi/7) == -tan(3*pi/7)
    assert tan(-11*pi/7) == tan(3*pi/7)

def test_tan():
    assert tan(nan) == nan

    assert tan.nargs == FiniteSet(1)
    assert tan(oo*I) == I
    assert tan(-oo*I) == -I

    assert tan(0) == 0

    assert tan(atan(x)) == x
    assert tan(asin(x)) == x / sqrt(1 - x**2)
    assert tan(acos(x)) == sqrt(1 - x**2) / x
    assert tan(acot(x)) == 1 / x
    assert tan(atan2(y, x)) == y/x

    assert tan(pi*I) == tanh(pi)*I
    assert tan(-pi*I) == -tanh(pi)*I
    assert tan(-2*I) == -tanh(2)*I

    assert tan(pi) == 0
    assert tan(-pi) == 0
    assert tan(2*pi) == 0
    assert tan(-2*pi) == 0
    assert tan(-3*10**73*pi) == 0

    assert tan(pi/2) == zoo
    assert tan(3*pi/2) == zoo

    assert tan(pi/3) == sqrt(3)
    assert tan(-2*pi/3) == sqrt(3)

    assert tan(pi/4) == S.One
    assert tan(-pi/4) == -S.One
    assert tan(17*pi/4) == S.One
    assert tan(-3*pi/4) == S.One

    assert tan(pi/6) == 1/sqrt(3)
    assert tan(-pi/6) == -1/sqrt(3)
    assert tan(7*pi/6) == 1/sqrt(3)
    assert tan(-5*pi/6) == 1/sqrt(3)

    assert tan(x*I) == tanh(x)*I

    assert tan(k*pi) == 0
    assert tan(17*k*pi) == 0

    assert tan(k*pi*I) == tanh(k*pi)*I

    assert tan(r).is_real is True

    assert tan(0, evaluate=False).is_algebraic
    assert tan(a).is_algebraic is None
    assert tan(na).is_algebraic is False

    assert tan(10*pi/7) == tan(3*pi/7)
    assert tan(11*pi/7) == -tan(3*pi/7)
    assert tan(-11*pi/7) == tan(3*pi/7)

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 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 splice(a, b, x0, width, x, dx):
    from sympy import tanh
    ax = a(x)
    bx = b(x)
    dax = symdiff(a)(x)
    dbx = symdiff(b)(x)
    f = ax + (bx-ax) * (1+tanh((x-x0)/width)) / 2
    df = (dax
          + (dbx-dax) * (1+tanh((x-x0)/width)) / 2
          + (bx - ax) * (1-tanh((x-x0)/width)**2) / (2*width)) * dx
    return f, df

def test_trigsimp1a():
    assert trigsimp(sin(2)**2*cos(3)*exp(2)/cos(2)**2) == tan(2)**2*cos(3)*exp(2)
    assert trigsimp(tan(2)**2*cos(3)*exp(2)*cos(2)**2) == sin(2)**2*cos(3)*exp(2)
    assert trigsimp(cot(2)*cos(3)*exp(2)*sin(2)) == cos(3)*exp(2)*cos(2)
    assert trigsimp(tan(2)*cos(3)*exp(2)/sin(2)) == cos(3)*exp(2)/cos(2)
    assert trigsimp(cot(2)*cos(3)*exp(2)/cos(2)) == cos(3)*exp(2)/sin(2)
    assert trigsimp(cot(2)*cos(3)*exp(2)*tan(2)) == cos(3)*exp(2)
    assert trigsimp(sinh(2)*cos(3)*exp(2)/cosh(2)) == tanh(2)*cos(3)*exp(2)
    assert trigsimp(tanh(2)*cos(3)*exp(2)*cosh(2)) == sinh(2)*cos(3)*exp(2)
    assert trigsimp(coth(2)*cos(3)*exp(2)*sinh(2)) == cosh(2)*cos(3)*exp(2)
    assert trigsimp(tanh(2)*cos(3)*exp(2)/sinh(2)) == cos(3)*exp(2)/cosh(2)
    assert trigsimp(coth(2)*cos(3)*exp(2)/cosh(2)) == cos(3)*exp(2)/sinh(2)
    assert trigsimp(coth(2)*cos(3)*exp(2)*tanh(2)) == cos(3)*exp(2)

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

    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

def test_cos_rewrite():
    assert cos(x).rewrite(exp) == exp(I * x) / 2 + exp(-I * x) / 2
    assert cos(x).rewrite(tan) == (1 - tan(x / 2) ** 2) / (1 + tan(x / 2) ** 2)
    assert cos(x).rewrite(cot) == -(1 - cot(x / 2) ** 2) / (1 + cot(x / 2) ** 2)
    assert cos(sinh(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sinh(3)).n()
    assert cos(cosh(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cosh(3)).n()
    assert cos(tanh(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tanh(3)).n()
    assert cos(coth(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, coth(3)).n()
    assert cos(sin(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sin(3)).n()
    assert cos(cos(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cos(3)).n()
    assert cos(tan(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tan(3)).n()
    assert cos(cot(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cot(3)).n()
    assert cos(log(x)).rewrite(Pow) == x ** I / 2 + x ** -I / 2

def test_sin_rewrite():
    assert sin(x).rewrite(exp) == -I * (exp(I * x) - exp(-I * x)) / 2
    assert sin(x).rewrite(tan) == 2 * tan(x / 2) / (1 + tan(x / 2) ** 2)
    assert sin(x).rewrite(cot) == 2 * cot(x / 2) / (1 + cot(x / 2) ** 2)
    assert sin(sinh(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sinh(3)).n()
    assert sin(cosh(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cosh(3)).n()
    assert sin(tanh(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tanh(3)).n()
    assert sin(coth(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, coth(3)).n()
    assert sin(sin(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sin(3)).n()
    assert sin(cos(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cos(3)).n()
    assert sin(tan(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tan(3)).n()
    assert sin(cot(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cot(3)).n()
    assert sin(log(x)).rewrite(Pow) == I * x ** -I / 2 - I * x ** I / 2

def test_cot_rewrite():
    x = Symbol('x')
    neg_exp, pos_exp = exp(-x*I), exp(x*I)
    assert cot(x).rewrite(exp) == I*(pos_exp+neg_exp)/(pos_exp-neg_exp)
    assert cot(x).rewrite(sin) == 2*sin(2*x)/sin(x)**2
    assert cot(x).rewrite(cos) == -cos(x)/cos(x + S.Pi/2)
    assert cot(x).rewrite(tan) == 1/tan(x)
    assert cot(sinh(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, sinh(3)).n()
    assert cot(cosh(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, cosh(3)).n()
    assert cot(tanh(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, tanh(3)).n()
    assert cot(coth(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, coth(3)).n()
    assert cot(sin(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, sin(3)).n()
    assert cot(tan(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, tan(3)).n()
    assert cot(log(x)).rewrite(Pow) == -I*(x**-I + x**I)/(x**-I - x**I)