Python sympy.asinh函数代码示例

def test_asinh_series():
    x = Symbol('x')
    assert asinh(x).series(x, 0, 8) == \
                x - x**3/6 + 3*x**5/40 - 5*x**7/112 + O(x**8)
    t5 = asinh(x).taylor_term(5, x)
    assert t5 == 3*x**5/40
    assert asinh(x).taylor_term(7, x, t5, 0) == -5*x**7/112

def test_heurisch_hyperbolic():
    assert heurisch(sinh(x), x) == cosh(x)
    assert heurisch(cosh(x), x) == sinh(x)

    assert heurisch(x*sinh(x), x) == x*cosh(x) - sinh(x)
    assert heurisch(x*cosh(x), x) == x*sinh(x) - cosh(x)

    assert heurisch(x*asinh(x/2), x) == x**2*asinh(x/2)/2 + asinh(x/2) - x*sqrt(4+x**2)/4

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_length():
    t = Symbol('t', real=True)

    c1 = Curve((t, 0), (t, 0, 1))
    assert c1.length == 1

    c2 = Curve((t, t), (t, 0, 1))
    assert c2.length == sqrt(2)

    c3 = Curve((t ** 2, t), (t, 2, 5))
    assert c3.length == -sqrt(17) - asinh(4) / 4 + asinh(10) / 4 + 5 * sqrt(101) / 2

def test_manualintegrate_inversetrig():
    # atan
    assert manualintegrate(exp(x) / (1 + exp(2*x)), x) == atan(exp(x))
    assert manualintegrate(1 / (4 + 9 * x**2), x) == atan(3 * x/2) / 6
    assert manualintegrate(1 / (16 + 16 * x**2), x) == atan(x) / 16
    assert manualintegrate(1 / (4 + x**2), x) == atan(x / 2) / 2
    assert manualintegrate(1 / (1 + 4 * x**2), x) == atan(2*x) / 2
    assert manualintegrate(1/(a + b*x**2), x) == \
        Piecewise(((sqrt(a/b)*atan(x*sqrt(b/a))/a), And(a > 0, b > 0)))
    assert manualintegrate(1/(4 + b*x**2), x) == \
        Piecewise((sqrt(1/b)*atan(sqrt(b)*x/2)/2, b > 0))
    assert manualintegrate(1/(a + 4*x**2), x) == \
        Piecewise((atan(2*x*sqrt(1/a))/(2*sqrt(a)), a > 0))
    assert manualintegrate(1/(4 + 4*x**2), x) == atan(x) / 4

    # asin
    assert manualintegrate(1/sqrt(1-x**2), x) == asin(x)
    assert manualintegrate(1/sqrt(4-4*x**2), x) == asin(x)/2
    assert manualintegrate(3/sqrt(1-9*x**2), x) == asin(3*x)
    assert manualintegrate(1/sqrt(4-9*x**2), x) == asin(3*x/2)/3

    # asinh
    assert manualintegrate(1/sqrt(x**2 + 1), x) == \
        asinh(x)
    assert manualintegrate(1/sqrt(x**2 + 4), x) == \
        asinh(x/2)
    assert manualintegrate(1/sqrt(4*x**2 + 4), x) == \
        asinh(x)/2
    assert manualintegrate(1/sqrt(4*x**2 + 1), x) == \
        asinh(2*x)/2
    assert manualintegrate(1/sqrt(a*x**2 + 1), x) == \
        Piecewise((sqrt(-1/a)*asin(x*sqrt(-a)), a < 0), (sqrt(1/a)*asinh(sqrt(a)*x), a > 0))
    assert manualintegrate(1/sqrt(a + x**2), x) == \
        Piecewise((asinh(x*sqrt(1/a)), a > 0), (acosh(x*sqrt(-1/a)), a < 0))

    # acosh
    assert manualintegrate(1/sqrt(x**2 - 1), x) == \
        acosh(x)
    assert manualintegrate(1/sqrt(x**2 - 4), x) == \
        acosh(x/2)
    assert manualintegrate(1/sqrt(4*x**2 - 4), x) == \
        acosh(x)/2
    assert manualintegrate(1/sqrt(9*x**2 - 1), x) == \
        acosh(3*x)/3
    assert manualintegrate(1/sqrt(a*x**2 - 4), x) == \
        Piecewise((sqrt(1/a)*acosh(sqrt(a)*x/2), a > 0))
    assert manualintegrate(1/sqrt(-a + 4*x**2), x) == \
        Piecewise((asinh(2*x*sqrt(-1/a))/2, -a > 0), (acosh(2*x*sqrt(1/a))/2, -a < 0))

    # piecewise
    assert manualintegrate(1/sqrt(a-b*x**2), x) == \
        Piecewise((sqrt(a/b)*asin(x*sqrt(b/a))/sqrt(a), And(-b < 0, a > 0)),
                  (sqrt(-a/b)*asinh(x*sqrt(-b/a))/sqrt(a), And(-b > 0, a > 0)),
                  (sqrt(a/b)*acosh(x*sqrt(b/a))/sqrt(-a), And(-b > 0, a < 0)))
    assert manualintegrate(1/sqrt(a + b*x**2), x) == \
        Piecewise((sqrt(-a/b)*asin(x*sqrt(-b/a))/sqrt(a), And(a > 0, b < 0)),
                  (sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(a), And(a > 0, b > 0)),
                  (sqrt(-a/b)*acosh(x*sqrt(-b/a))/sqrt(-a), And(a < 0, b > 0)))

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 eta_fil(self, x, V_app, apprx=(0, 0, 0, 0)):
        m_eff = self.m_r * const.electron_mass

        mpmath.mp.dps = 20
        x0 = Symbol('x0')  # eta_fil
        x1 = Symbol('x1')  # eta_ac
        x2 = Symbol('x2')  # eta_hop
        x3 = Symbol('x3')  # V_tunnel

        f0 = const.Boltzmann * self.T / (1 - self.alpha) / const.elementary_charge / self.z * \
             ln(self.A_fil/self.A_ac*(exp(- self.alpha * const.elementary_charge * self.z / const.Boltzmann / self.T * x0) - 1) + 1) - x1# eta_ac = f(eta_fil) x1 = f(x0)
        f1 = x*2*const.Boltzmann*self.T/self.a/self.z/const.elementary_charge*\
             asinh(self.j_0et/self.j_0hop*(exp(- self.alpha * const.elementary_charge * self.z / const.Boltzmann / self.T * x0) - 1)) - x2# eta_hop = f(eta_fil)
        f2 = x1 - x0 + x2 - x3

        f3 = -V_app + ((self.C * 3 * sqrt(2 * m_eff * ((4+x3/2)*const.elementary_charge)) / 2 / x * (const.elementary_charge / const.Planck)**2 * \
             exp(- 4 * const.pi * x / const.Planck * sqrt(2 * m_eff * ((4+x3/2)*const.elementary_charge))) * self.A_fil*x3)
                       + (self.j_0et*self.A_fil*(exp(-self.alpha*const.elementary_charge*self.z*x0/const.Boltzmann/self.T) - 1))) * (self.R_el + self.R_S + self.rho_fil*(self.L - x) / self.A_fil) \
             + x3

        eta_fil, eta_ac, eta_hop, V_tunnel = nsolve((f0, f1, f2, f3), [x0, x1, x2, x3], apprx)
        eta_fil = np.real(np.complex128(eta_fil))
        eta_ac = np.real(np.complex128(eta_ac))
        eta_hop = np.real(np.complex128(eta_hop))
        V_tunnel = np.real(np.complex128(V_tunnel))
        current = ((self.C * 3 * sqrt(2 * m_eff * ((4+V_tunnel)*const.elementary_charge)) / 2 / x * (const.elementary_charge / const.Planck)**2 * \
            exp(- 4 * const.pi * x / const.Planck * sqrt(2 * m_eff * ((4+V_tunnel)*const.elementary_charge))) * self.A_fil*V_tunnel)
                       + (self.j_0et*self.A_fil*(exp(-self.alpha*const.elementary_charge*self.z*eta_fil/const.Boltzmann/self.T) - 1)))
        print(eta_fil, eta_ac, eta_hop, V_tunnel)
        # print(eta_ac - eta_fil + eta_hop - V_tunnel)
        return eta_fil, eta_ac, eta_hop, V_tunnel, current

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

    assert asin(nan) == nan

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

    # Note: asin(-x) = - asin(x)
    assert asin(0)  == 0
    assert asin(1)  == pi/2
    assert asin(-1)  == -pi/2
    assert asin(sqrt(3)/2) == pi/3
    assert asin(-sqrt(3)/2) == -pi/3
    assert asin(sqrt(2)/2) == pi/4
    assert asin(-sqrt(2)/2) == -pi/4
    assert asin(sqrt((5-sqrt(5))/8)) == pi/5
    assert asin(-sqrt((5-sqrt(5))/8)) == -pi/5
    assert asin(Rational(1,2)) == pi/6
    assert asin(-Rational(1,2)) == -pi/6
    assert asin((sqrt(2-sqrt(2)))/2) == pi/8
    assert asin(-(sqrt(2-sqrt(2)))/2) == -pi/8
    assert asin((sqrt(5)-1)/4) == pi/10
    assert asin(-(sqrt(5)-1)/4) == -pi/10
    assert asin((sqrt(3)-1)/sqrt(2**3)) == pi/12
    assert asin(-(sqrt(3)-1)/sqrt(2**3)) == -pi/12

    assert asin(x).diff(x) ==  1/sqrt(1-x**2)

    assert asin(0.2).is_real == True
    assert asin(-2).is_real == False

    assert asin(-2*I) == -I*asinh(2)

def test_issue_1304():
    z = Symbol("z", positive=True)
    assert integrate(sqrt(x ** 2 + z ** 2), x) == z ** 2 * asinh(x / z) / 2 + x * (x ** 2 + z ** 2) ** (S(1) / 2) / 2
    assert integrate(sqrt(x ** 2 - z ** 2), x) == -z ** 2 * acosh(x / z) / 2 + x * (x ** 2 - z ** 2) ** (S(1) / 2) / 2
    assert (
        integrate(sqrt(-x ** 2 - 4), x)
        == -2 * atan(x / (-4 - x ** 2) ** (S(1) / 2)) + x * (-4 - x ** 2) ** (S(1) / 2) / 2
    )

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 test_hyperbolic():
    x = Symbol("x")
    assert sinh(x).nseries(x, 0, 6) == x + x**3/6 + x**5/120 + O(x**6)
    assert cosh(x).nseries(x, 0, 5) == 1 + x**2/2 + x**4/24 + O(x**5)
    assert tanh(x).nseries(x, 0, 6) == x - x**3/3 + 2*x**5/15 + O(x**6)
    assert coth(x).nseries(x, 0, 6) == 1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**6)
    assert asinh(x).nseries(x, 0, 6) == x - x**3/6 + 3*x**5/40 + O(x**6)
    assert acosh(x).nseries(x, 0, 6) == pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**6)
    assert atanh(x).nseries(x, 0, 6) == x + x**3/3 + x**5/5 + O(x**6)
    assert acoth(x).nseries(x, 0, 6) == x + x**3/3 + x**5/5 + pi*I/2 + O(x**6)

def test_heurisch_hacking():
    assert heurisch(sqrt(1 + 7*x**2), x, hints=[]) == \
        x*sqrt(1+7*x**2)/2 + sqrt(7)*asinh(sqrt(7)*x)/14
    assert heurisch(sqrt(1 - 7*x**2), x, hints=[]) == \
        x*sqrt(1-7*x**2)/2 + sqrt(7)*asin(sqrt(7)*x)/14

    assert heurisch(1/sqrt(1 + 7*x**2), x, hints=[]) == \
        sqrt(7)*asinh(sqrt(7)*x)/7
    assert heurisch(1/sqrt(1 - 7*x**2), x, hints=[]) == \
        sqrt(7)*asin(sqrt(7)*x)/7

    assert heurisch(exp(-7*x**2),x,hints=[]) == \
        sqrt(7*pi)*erf(sqrt(7)*x)/14

    assert heurisch(1/sqrt(9 - 4*x**2), x, hints=[]) == \
        asin(2*x/3)/2

    assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == \
        asinh(2*x/3)/2

def test_inverses():
    x = Symbol('x')
    assert sinh(x).inverse() == asinh
    raises(AttributeError, lambda: cosh(x).inverse())
    assert tanh(x).inverse() == atanh
    assert coth(x).inverse() == acoth
    assert asinh(x).inverse() == sinh
    assert acosh(x).inverse() == cosh
    assert atanh(x).inverse() == tanh
    assert acoth(x).inverse() == coth

def test_conv12b():
    x = sympy.Symbol("x")
    y = sympy.Symbol("y")
    assert sympify(sympy.sinh(x/3)) == sinh(Symbol("x") / 3)
    assert sympify(sympy.cosh(x/3)) == cosh(Symbol("x") / 3)
    assert sympify(sympy.tanh(x/3)) == tanh(Symbol("x") / 3)
    assert sympify(sympy.coth(x/3)) == coth(Symbol("x") / 3)
    assert sympify(sympy.asinh(x/3)) == asinh(Symbol("x") / 3)
    assert sympify(sympy.acosh(x/3)) == acosh(Symbol("x") / 3)
    assert sympify(sympy.atanh(x/3)) == atanh(Symbol("x") / 3)
    assert sympify(sympy.acoth(x/3)) == acoth(Symbol("x") / 3)

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