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_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*(4+x**2)**Rational(1,2)/4
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 unchanged(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) is 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
assert acsch(nan) is nan
# 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_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 balanceFun(z):
c = float(z[0])
return [
-c *
sympy.cosh(0.5 * l / c - sympy.asinh(0.5 * h /
(c * sympy.sinh(0.5 * l / c))) -
0.5 * l / c) + c *
sympy.cosh(-sympy.asinh(0.5 * h /
(c * sympy.sinh(0.5 * l / c))) - 0.5 * l / c) -
(h / 2 + n)
]
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_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_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_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 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
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_trig_functions(self, printer, x):
# Trig functions
assert printer.doprint(sp.acos(x)) == 'acos(x)'
assert printer.doprint(sp.acosh(x)) == 'acosh(x)'
assert printer.doprint(sp.asin(x)) == 'asin(x)'
assert printer.doprint(sp.asinh(x)) == 'asinh(x)'
assert printer.doprint(sp.atan(x)) == 'atan(x)'
assert printer.doprint(sp.atanh(x)) == 'atanh(x)'
assert printer.doprint(sp.ceiling(x)) == 'ceil(x)'
assert printer.doprint(sp.cos(x)) == 'cos(x)'
assert printer.doprint(sp.cosh(x)) == 'cosh(x)'
assert printer.doprint(sp.exp(x)) == 'exp(x)'
assert printer.doprint(sp.factorial(x)) == 'factorial(x)'
assert printer.doprint(sp.floor(x)) == 'floor(x)'
assert printer.doprint(sp.log(x)) == 'log(x)'
assert printer.doprint(sp.sin(x)) == 'sin(x)'
assert printer.doprint(sp.sinh(x)) == 'sinh(x)'
assert printer.doprint(sp.tan(x)) == 'tan(x)'
assert printer.doprint(sp.tanh(x)) == 'tanh(x)'
# extra trig functions
assert printer.doprint(sp.sec(x)) == '1 / cos(x)'
assert printer.doprint(sp.csc(x)) == '1 / sin(x)'
assert printer.doprint(sp.cot(x)) == '1 / tan(x)'
assert printer.doprint(sp.asec(x)) == 'acos(1 / x)'
assert printer.doprint(sp.acsc(x)) == 'asin(1 / x)'
assert printer.doprint(sp.acot(x)) == 'atan(1 / x)'
assert printer.doprint(sp.sech(x)) == '1 / cosh(x)'
assert printer.doprint(sp.csch(x)) == '1 / sinh(x)'
assert printer.doprint(sp.coth(x)) == '1 / tanh(x)'
assert printer.doprint(sp.asech(x)) == 'acosh(1 / x)'
assert printer.doprint(sp.acsch(x)) == 'asinh(1 / x)'
assert printer.doprint(sp.acoth(x)) == 'atanh(1 / 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_math_functions():
dort('sin(37)', sympy.sin(37))
dort('cos(38)', sympy.cos(38))
dort('tan(38)', sympy.tan(38))
dort('sec(39)', sympy.sec(39))
dort('csc(40)', sympy.csc(40))
dort('cot(41)', sympy.cot(41))
dort('asin(42)', sympy.asin(42))
dort('acos(43)', sympy.acos(43))
dort('atan(44)', sympy.atan(44))
dort('asec(45)', sympy.asec(45))
dort('acsc(46)', sympy.acsc(46))
dort('acot(47)', sympy.acot(47))
dort('sind(37)', sympy.sin(37 * sympy.pi / sympy.Number(180)))
dort('cosd(38)', sympy.cos(38 * sympy.pi / sympy.Number(180)))
dort('tand(38)', sympy.tan(38 * sympy.pi / sympy.Number(180)))
dort('secd(39)', sympy.sec(39 * sympy.pi / sympy.Number(180)))
dort('cscd(40)', sympy.csc(40 * sympy.pi / sympy.Number(180)))
dort('cotd(41)', sympy.cot(41 * sympy.pi / sympy.Number(180)))
dort('asind(42)', sympy.asin(42) * sympy.Number(180) / sympy.pi)
dort('acosd(43)', sympy.acos(43) * sympy.Number(180) / sympy.pi)
dort('atand(44)', sympy.atan(44) * sympy.Number(180) / sympy.pi)
dort('asecd(45)', sympy.asec(45) * sympy.Number(180) / sympy.pi)
dort('acscd(46)', sympy.acsc(46) * sympy.Number(180) / sympy.pi)
dort('acotd(47)', sympy.acot(47) * sympy.Number(180) / sympy.pi)
dort('sinh(4)', sympy.sinh(4))
dort('cosh(5)', sympy.cosh(5))
dort('tanh(6)', sympy.tanh(6))
dort('asinh(4)', sympy.asinh(4))
dort('acosh(5)', sympy.acosh(5))
dort('atanh(6)', sympy.atanh(6))
dort('int(E)', int(sympy.E))
dort('int(-E)', int(-sympy.E))
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 asinh(self, value):
"""
Verilen değeri kullanarak, sumpy.asinh döndürür
:param value: değer
:return: sympy.asinh
"""
return sp.asinh(value_checker(value))
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 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_explicit_latex(self):
test_cases = [
('0x', Mul(0, x, evaluate=False)),
('0 = 0', Eq(0, 0)),
('0 = 1', Eq(0, 1)),
('10x = 10x', Eq(10 * x, 10 * x)),
(r'\sin ^ { - 1 } ( x )', asin(x)),
(r'\cos^{-1}x', acos(x)),
(r'\tan^ {-1 }x', atan(x)),
(r'\cot^ {-1 }x', acot(x)),
(r'\sinh^{-1}(x)', asinh(x)),
(r'\sqrt{-1}', I),
(r'\exp{x}', exp(x)),
(r'\exp(x)', exp(x)),
('e^{x}', E**x),
]
failed_tests = []
for s_l_expr, expr in test_cases:
l_expr = process_sympy(s_l_expr)
equiv = equivalent(expr, l_expr)
if not equiv:
print '%s %s' % (s_l_expr, 'PASSED' if equiv else 'FAILED')
failed_tests.append((s_l_expr, l_expr))
print 'sympy: %s\nlatex: %s\n' % (expr, l_expr)
if failed_tests:
print len(failed_tests), 'failed test cases'
assert len(failed_tests) == 0
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
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_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(x*Rational(2, 3))/2
assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == \
asinh(x*Rational(2, 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_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_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_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_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_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
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_issue_4403():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z', positive=True)
assert integrate(sqrt(x**2 + z**2), x) == \
z**2*asinh(x/z)/2 + x*sqrt(x**2 + z**2)/2
assert integrate(sqrt(x**2 - z**2), x) == \
-z**2*acosh(x/z)/2 + x*sqrt(x**2 - z**2)/2
x = Symbol('x', real=True)
y = Symbol('y', nonzero=True, real=True)
assert integrate(1/(x**2 + y**2)**S('3/2'), x) == \
1/(y**2*sqrt(1 + y**2/x**2))
def test_issue_4403():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z', positive=True)
assert integrate(sqrt(x**2 + z**2), x) == \
z**2*asinh(x/z)/2 + x*sqrt(x**2 + z**2)/2
assert integrate(sqrt(x**2 - z**2), x) == \
-z**2*acosh(x/z)/2 + x*sqrt(x**2 - z**2)/2
x = Symbol('x', real=True)
y = Symbol('y', positive=True)
assert integrate(1/(x**2 + y**2)**S('3/2'), x) == \
x/(y**2*sqrt(x**2 + y**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 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)
assert asech(x).diff(x) == -1 / (x * sqrt(1 - x**2))
assert acsch(x).diff(x) == -1 / (x**2 * sqrt(1 + x**(-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 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)
assert asech(x).diff(x) == -1/(x*sqrt(1 - x**2))
assert acsch(x).diff(x) == -1/(x**2*sqrt(1 + x**(-2)))
def test_to_equation(self):
a = pybamm.Symbol("a", domain="test")
# Test _sympy_operator
with self.assertRaises(NotImplementedError):
pybamm.Arctan(a).to_equation()
# Test print_name
func = pybamm.Arcsinh(a)
func.print_name = "test"
self.assertEqual(func.to_equation(), sympy.symbols("test"))
# Test Arcsinh
self.assertEqual(pybamm.Arcsinh(a).to_equation(), sympy.asinh(a))
def test_asinh_noimpl():
assert asinh(I * (sqrt(3) - 1) / (2**(3 / 2))) == pi * I / 12
assert asinh(-I * (sqrt(3) - 1) / (2**(3 / 2))) == -pi * I / 12
assert asinh(I * (sqrt(5) - 1) / 4) == pi * I / 10
assert asinh(-I * (sqrt(5) - 1) / 4) == -pi * I / 10
assert asinh(I * (sqrt(5) + 1) / 4) == 3 * pi * I / 10
assert asinh(-I * (sqrt(5) + 1) / 4) == -3 * pi * I / 10
def test_asinh_noimpl():
assert asinh(I *(sqrt(3) - 1)/(2**(3/2))) == pi*I/12
assert asinh(-I *(sqrt(3) - 1)/(2**(3/2))) == -pi*I/12
assert asinh(I*(sqrt(5)-1)/4) == pi*I/10
assert asinh(-I*(sqrt(5)-1)/4) == -pi*I/10
assert asinh(I*(sqrt(5)+1)/4) == 3*pi*I/10
assert asinh(-I*(sqrt(5)+1)/4) == -3*pi*I/10
def test_integrate():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 2, 3))
q = '0.166270406994788'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr()
q = 1 - cos(x)
assert p == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, 3))
q = 1 - cos(3)
assert p == q
p = expr_to_holonomic(sin(x) / x, x0=1).integrate((x, 1, 2))
q = '0.659329913368450'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 1, 0))
q = '-0.423690480850035'
assert sstr(p) == q
p = expr_to_holonomic(sin(x) / x)
assert p.integrate(x).to_expr() == Si(x)
assert p.integrate((x, 0, 2)) == Si(2)
p = expr_to_holonomic(sin(x)**2 / x)
q = p.to_expr()
assert p.integrate(x).to_expr() == q.integrate((x, 0, x))
assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1))
assert expr_to_holonomic(1 / x, x0=1).integrate(x).to_expr() == log(x)
p = expr_to_holonomic((x + 1)**3 * exp(-x), x0=-1).integrate(x).to_expr()
q = (-x**3 - 6 * x**2 - 15 * x + 6 * exp(x + 1) - 16) * exp(-x)
assert p == q
p = expr_to_holonomic(cos(x)**2 / x**2, y0={
-2: [1, 0, -1]
}).integrate(x).to_expr()
q = -Si(2 * x) - cos(x)**2 / x
assert p == q
p = expr_to_holonomic(sqrt(x**2 + x)).integrate(x).to_expr()
q = (x**Rational(3, 2) * (2 * x**2 + 3 * x + 1) -
x * sqrt(x + 1) * asinh(sqrt(x))) / (4 * x * sqrt(x + 1))
assert p == q
p = expr_to_holonomic(sqrt(x**2 + 1)).integrate(x).to_expr()
q = (sqrt(x**2 + 1)).integrate(x)
assert (p - q).simplify() == 0
p = expr_to_holonomic(1 / x**2, y0={-2: [1, 0, 0]})
r = expr_to_holonomic(1 / x**2, lenics=3)
assert p == r
q = expr_to_holonomic(cos(x)**2)
assert (r * q).integrate(x).to_expr() == -Si(2 * x) - cos(x)**2 / x
def test_integrate():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 2, 3))
q = '0.166270406994788'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr()
q = 1 - cos(x)
assert p == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, 3))
q = 1 - cos(3)
assert p == q
p = expr_to_holonomic(sin(x)/x, x0=1).integrate((x, 1, 2))
q = '0.659329913368450'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 1, 0))
q = '-0.423690480850035'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)/x)
assert p.integrate(x).to_expr() == Si(x)
assert p.integrate((x, 0, 2)) == Si(2)
p = expr_to_holonomic(sin(x)**2/x)
q = p.to_expr()
assert p.integrate(x).to_expr() == q.integrate((x, 0, x))
assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1))
assert expr_to_holonomic(1/x, x0=1).integrate(x).to_expr() == log(x)
p = expr_to_holonomic((x + 1)**3*exp(-x), x0=-1).integrate(x).to_expr()
q = (-x**3 - 6*x**2 - 15*x + 6*exp(x + 1) - 16)*exp(-x)
assert p == q
p = expr_to_holonomic(cos(x)**2/x**2, y0={-2: [1, 0, -1]}).integrate(x).to_expr()
q = -Si(2*x) - cos(x)**2/x
assert p == q
p = expr_to_holonomic(sqrt(x**2+x)).integrate(x).to_expr()
q = (x**(3/2)*(2*x**2 + 3*x + 1) - x*sqrt(x + 1)*asinh(sqrt(x)))/(4*x*sqrt(x + 1))
assert p == q
p = expr_to_holonomic(sqrt(x**2+1)).integrate(x).to_expr()
q = (sqrt(x**2+1)).integrate(x)
assert (p-q).simplify() == 0
p = expr_to_holonomic(1/x**2, y0={-2:[1, 0, 0]})
r = expr_to_holonomic(1/x**2, lenics=3)
assert p == r
q = expr_to_holonomic(cos(x)**2)
assert (r*q).integrate(x).to_expr() == -Si(2*x) - cos(x)**2/x
def test_asin():
x = Symbol('x')
assert asin(nan) == nan
assert asin(oo) == -I * oo
assert asin(-oo) == I * oo
assert asin(0) == 0
assert asin(Rational(1, 2)) == pi / 6
assert asin(1) == pi / 2
assert asin(sqrt(3) / 2) == pi / 3
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_asin():
x = Symbol('x')
assert asin(nan) == nan
assert asin(oo) == -I*oo
assert asin(-oo) == I*oo
assert asin(0) == 0
assert asin(Rational(1,2)) == pi/6
assert asin(1) == pi/2
assert asin(sqrt(3)/2) == pi/3
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_conv12():
x = Symbol("x")
y = Symbol("y")
assert sinh(x/3) == sinh(sympy.Symbol("x") / 3)
assert cosh(x/3) == cosh(sympy.Symbol("x") / 3)
assert tanh(x/3) == tanh(sympy.Symbol("x") / 3)
assert coth(x/3) == coth(sympy.Symbol("x") / 3)
assert asinh(x/3) == asinh(sympy.Symbol("x") / 3)
assert acosh(x/3) == acosh(sympy.Symbol("x") / 3)
assert atanh(x/3) == atanh(sympy.Symbol("x") / 3)
assert acoth(x/3) == acoth(sympy.Symbol("x") / 3)
assert sinh(x/3)._sympy_() == sympy.sinh(sympy.Symbol("x") / 3)
assert cosh(x/3)._sympy_() == sympy.cosh(sympy.Symbol("x") / 3)
assert tanh(x/3)._sympy_() == sympy.tanh(sympy.Symbol("x") / 3)
assert coth(x/3)._sympy_() == sympy.coth(sympy.Symbol("x") / 3)
assert asinh(x/3)._sympy_() == sympy.asinh(sympy.Symbol("x") / 3)
assert acosh(x/3)._sympy_() == sympy.acosh(sympy.Symbol("x") / 3)
assert atanh(x/3)._sympy_() == sympy.atanh(sympy.Symbol("x") / 3)
assert acoth(x/3)._sympy_() == sympy.acoth(sympy.Symbol("x") / 3)
def test_conv12():
x = Symbol("x")
y = Symbol("y")
assert sinh(x / 3) == sinh(sympy.Symbol("x") / 3)
assert cosh(x / 3) == cosh(sympy.Symbol("x") / 3)
assert tanh(x / 3) == tanh(sympy.Symbol("x") / 3)
assert coth(x / 3) == coth(sympy.Symbol("x") / 3)
assert asinh(x / 3) == asinh(sympy.Symbol("x") / 3)
assert acosh(x / 3) == acosh(sympy.Symbol("x") / 3)
assert atanh(x / 3) == atanh(sympy.Symbol("x") / 3)
assert acoth(x / 3) == acoth(sympy.Symbol("x") / 3)
assert sinh(x / 3)._sympy_() == sympy.sinh(sympy.Symbol("x") / 3)
assert cosh(x / 3)._sympy_() == sympy.cosh(sympy.Symbol("x") / 3)
assert tanh(x / 3)._sympy_() == sympy.tanh(sympy.Symbol("x") / 3)
assert coth(x / 3)._sympy_() == sympy.coth(sympy.Symbol("x") / 3)
assert asinh(x / 3)._sympy_() == sympy.asinh(sympy.Symbol("x") / 3)
assert acosh(x / 3)._sympy_() == sympy.acosh(sympy.Symbol("x") / 3)
assert atanh(x / 3)._sympy_() == sympy.atanh(sympy.Symbol("x") / 3)
assert acoth(x / 3)._sympy_() == sympy.acoth(sympy.Symbol("x") / 3)
def test_bng_printer():
# Constants
assert _bng_print(sympy.pi) == '_pi'
assert _bng_print(sympy.E) == '_e'
x, y = sympy.symbols('x y')
# Binary functions
assert _bng_print(sympy.sympify('x & y')) == 'x && y'
assert _bng_print(sympy.sympify('x | y')) == 'x || y'
# Trig functions
assert _bng_print(sympy.sin(x)) == 'sin(x)'
assert _bng_print(sympy.cos(x)) == 'cos(x)'
assert _bng_print(sympy.tan(x)) == 'tan(x)'
assert _bng_print(sympy.asin(x)) == 'asin(x)'
assert _bng_print(sympy.acos(x)) == 'acos(x)'
assert _bng_print(sympy.atan(x)) == 'atan(x)'
assert _bng_print(sympy.sinh(x)) == 'sinh(x)'
assert _bng_print(sympy.cosh(x)) == 'cosh(x)'
assert _bng_print(sympy.tanh(x)) == 'tanh(x)'
assert _bng_print(sympy.asinh(x)) == 'asinh(x)'
assert _bng_print(sympy.acosh(x)) == 'acosh(x)'
assert _bng_print(sympy.atanh(x)) == 'atanh(x)'
# Logs and powers
assert _bng_print(sympy.log(x)) == 'ln(x)'
assert _bng_print(sympy.exp(x)) == 'exp(x)'
assert _bng_print(sympy.sqrt(x)) == 'sqrt(x)'
# Rounding
assert _bng_print(sympy.Abs(x)) == 'abs(x)'
assert _bng_print(sympy.floor(x)) == 'rint(x - 0.5)'
assert _bng_print(sympy.ceiling(x)) == '(rint(x + 1) - 1)'
# Min/max
assert _bng_print(sympy.Min(x, y)) == 'min(x, y)'
assert _bng_print(sympy.Max(x, y)) == 'max(x, y)'
def test_bng_printer():
# Constants
assert _bng_print(sympy.pi) == '_pi'
assert _bng_print(sympy.E) == '_e'
x, y = sympy.symbols('x y')
# Binary functions
assert _bng_print(sympy.sympify('x & y')) == 'x && y'
assert _bng_print(sympy.sympify('x | y')) == 'x || y'
# Trig functions
assert _bng_print(sympy.sin(x)) == 'sin(x)'
assert _bng_print(sympy.cos(x)) == 'cos(x)'
assert _bng_print(sympy.tan(x)) == 'tan(x)'
assert _bng_print(sympy.asin(x)) == 'asin(x)'
assert _bng_print(sympy.acos(x)) == 'acos(x)'
assert _bng_print(sympy.atan(x)) == 'atan(x)'
assert _bng_print(sympy.sinh(x)) == 'sinh(x)'
assert _bng_print(sympy.cosh(x)) == 'cosh(x)'
assert _bng_print(sympy.tanh(x)) == 'tanh(x)'
assert _bng_print(sympy.asinh(x)) == 'asinh(x)'
assert _bng_print(sympy.acosh(x)) == 'acosh(x)'
assert _bng_print(sympy.atanh(x)) == 'atanh(x)'
# Logs and powers
assert _bng_print(sympy.log(x)) == 'ln(x)'
assert _bng_print(sympy.exp(x)) == 'exp(x)'
assert _bng_print(sympy.sqrt(x)) == 'sqrt(x)'
# Rounding
assert _bng_print(sympy.Abs(x)) == 'abs(x)'
assert _bng_print(sympy.floor(x)) == 'rint(x - 0.5)'
assert _bng_print(sympy.ceiling(x)) == '(rint(x + 1) - 1)'
# Min/max
assert _bng_print(sympy.Min(x, y)) == 'min(x, y)'
assert _bng_print(sympy.Max(x, y)) == 'max(x, y)'
def test_asin():
assert asin(nan) == nan
assert asin.nargs == FiniteSet(1)
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 is True
assert asin(-2).is_real is False
assert asin(r).is_real is None
assert asin(-2*I) == -I*asinh(2)
assert asin(Rational(1, 7), evaluate=False).is_positive is True
assert asin(Rational(-1, 7), evaluate=False).is_positive is False
assert asin(p).is_positive is None
def test_presentation_mathml_trig():
mml = mpp._print(sin(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'sin'
mml = mpp._print(cos(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'cos'
mml = mpp._print(tan(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'tan'
mml = mpp._print(asin(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arcsin'
mml = mpp._print(acos(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arccos'
mml = mpp._print(atan(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arctan'
mml = mpp._print(sinh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'sinh'
mml = mpp._print(cosh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'cosh'
mml = mpp._print(tanh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'tanh'
mml = mpp._print(asinh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arcsinh'
mml = mpp._print(atanh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arctanh'
mml = mpp._print(acosh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arccosh'
def test_mathml_trig():
mml = mp._print(sin(x))
assert mml.childNodes[0].nodeName == "sin"
mml = mp._print(cos(x))
assert mml.childNodes[0].nodeName == "cos"
mml = mp._print(tan(x))
assert mml.childNodes[0].nodeName == "tan"
mml = mp._print(asin(x))
assert mml.childNodes[0].nodeName == "arcsin"
mml = mp._print(acos(x))
assert mml.childNodes[0].nodeName == "arccos"
mml = mp._print(atan(x))
assert mml.childNodes[0].nodeName == "arctan"
mml = mp._print(sinh(x))
assert mml.childNodes[0].nodeName == "sinh"
mml = mp._print(cosh(x))
assert mml.childNodes[0].nodeName == "cosh"
mml = mp._print(tanh(x))
assert mml.childNodes[0].nodeName == "tanh"
mml = mp._print(asinh(x))
assert mml.childNodes[0].nodeName == "arcsinh"
mml = mp._print(atanh(x))
assert mml.childNodes[0].nodeName == "arctanh"
mml = mp._print(acosh(x))
assert mml.childNodes[0].nodeName == "arccosh"
def test_mathml_trig():
mml = mp._print(sin(x))
assert mml.childNodes[0].nodeName == 'sin'
mml = mp._print(cos(x))
assert mml.childNodes[0].nodeName == 'cos'
mml = mp._print(tan(x))
assert mml.childNodes[0].nodeName == 'tan'
mml = mp._print(asin(x))
assert mml.childNodes[0].nodeName == 'arcsin'
mml = mp._print(acos(x))
assert mml.childNodes[0].nodeName == 'arccos'
mml = mp._print(atan(x))
assert mml.childNodes[0].nodeName == 'arctan'
mml = mp._print(sinh(x))
assert mml.childNodes[0].nodeName == 'sinh'
mml = mp._print(cosh(x))
assert mml.childNodes[0].nodeName == 'cosh'
mml = mp._print(tanh(x))
assert mml.childNodes[0].nodeName == 'tanh'
mml = mp._print(asinh(x))
assert mml.childNodes[0].nodeName == 'arcsinh'
mml = mp._print(atanh(x))
assert mml.childNodes[0].nodeName == 'arctanh'
mml = mp._print(acosh(x))
assert mml.childNodes[0].nodeName == 'arccosh'
def test_asinh():
x, y = symbols('xy')
assert asinh(x) == asinh(x)
assert asinh(-x) == -asinh(x)
assert asinh(nan) == nan
assert asinh( 0) == 0
assert asinh(+1) == log(sqrt(2)+1)
assert asinh(-1) == log(sqrt(2)-1)
assert asinh(I) == pi*I/2
assert asinh(-I) == -pi*I/2
assert asinh(I/2) == pi*I/6
assert asinh(-I/2) == -pi*I/6
assert asinh(oo) == oo
assert asinh(-oo) == -oo
assert asinh(I*oo) == oo
assert asinh(-I *oo) == -oo
def test_issue_4422():
assert integrate(1/sqrt(16 + 4*x**2), x) == asinh(x/2) / 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*sqrt(x**2 + z**2)/2
assert integrate(sqrt(x**2 - z**2), x) == \
-z**2*acosh(x/z)/2 + x*sqrt(x**2 - z**2)/2
def test_issue1037():
assert cosh(asinh(Integer(3)/2)) == sqrt(Integer(13)/4)