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_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
assert heurisch(li(x), x, hints=[]) == x*li(x) - Ei(2*log(x))
def test_inverses():
assert sinh(x).inverse() == asinh
pytest.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_hyperbolic():
assert sinh(x).nseries(x, n=6) == x + x**3/6 + x**5/120 + O(x**7)
assert cosh(x).nseries(x, n=5) == 1 + x**2/2 + x**4/24 + O(x**6)
assert tanh(x).nseries(x, n=6) == x - x**3/3 + 2*x**5/15 + O(x**7)
assert coth(x).nseries(x, n=6) == \
1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**7)
assert asinh(x).nseries(x, n=6) == x - x**3/6 + 3*x**5/40 + O(x**7)
assert acosh(x).nseries(x, n=6) == \
pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**7)
assert atanh(x).nseries(x, n=6) == x + x**3/3 + x**5/5 + O(x**7)
assert acoth(x).nseries(x, n=6) == x + x**3/3 + x**5/5 + pi*I/2 + O(x**7)
def test_derivs():
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_sympyissue_4403():
x = Symbol('x')
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', extended_real=True)
y = Symbol('y', nonzero=True, extended_real=True)
assert integrate(1/(x**2 + y**2)**Rational(3, 2), x) == \
1/(y**2*sqrt(1 + y**2/x**2))
def test_simplifications():
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():
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_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(c).is_complex
assert asin(x).is_complex is None
assert asin(0.2).is_extended_real is True
assert asin(-2).is_extended_real is False
assert asin(r).is_extended_real is None
assert asin(0, evaluate=False).is_rational
assert asin(1, evaluate=False).is_rational is False
z = Symbol('z', zero=True)
rn = Symbol('rn', rational=True, nonzero=True)
assert asin(z).is_rational
assert asin(rn).is_rational is False
assert asin(x).is_rational 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
pytest.raises(ArgumentIndexError, lambda: asin(x).fdiff(2))
def test_numexprprinter():
p = NumExprPrinter()
M = MatrixSymbol('M', 1, 2)
pytest.raises(TypeError, lambda: p.doprint(M))
pytest.raises(TypeError, lambda: p.doprint([x, y]))
assert p.doprint(I) == "evaluate('1j')"
assert p.doprint(sin(x)) == "evaluate('sin(x)')"
assert p.doprint(asinh(x)) == "evaluate('arcsinh(x)')"
f = implemented_function('f', lambda x: 2 * x)
assert p.doprint(f(x)) == "evaluate('(2*x)')"
g = Function('g')
pytest.raises(TypeError, lambda: p.doprint(g(x)))
def test_numexprprinter():
p = NumExprPrinter()
M = MatrixSymbol('M', 1, 2)
pytest.raises(TypeError, lambda: p.doprint(M))
pytest.raises(TypeError, lambda: p.doprint([x, y]))
assert p.doprint(I) == "evaluate('1j')"
assert p.doprint(sin(x)) == "evaluate('sin(x)')"
assert p.doprint(asinh(x)) == "evaluate('arcsinh(x)')"
f = implemented_function('f', lambda x: 2*x)
assert p.doprint(f(x)) == "evaluate('(2*x)')"
g = Function('g')
pytest.raises(TypeError, lambda: p.doprint(g(x)))
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_RR():
# Make sure the algorithm does the right thing if the ring is RR. See
# issue sympy/sympy#8685.
assert heurisch(sqrt(1 + 0.25*x**2), x, hints=[]) == \
0.5*x*sqrt(0.25*x**2 + 1) + 1.0*asinh(0.5*x)
def test_sympyissue_4422():
assert integrate(1/sqrt(16 + 4*x**2), x) == asinh(x/2) / 2
def test_sympyissue_4422():
assert integrate(1 / sqrt(16 + 4 * x**2), x) == asinh(x / 2) / 2
def test_asinh():
x, y = symbols('x,y')
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
assert asinh(zoo) == zoo
assert asinh(I * (sqrt(3) - 1) / 2**Rational(3, 2)) == pi * I / 12
assert asinh(-I * (sqrt(3) - 1) / 2**Rational(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_sympyissue_4136():
assert cosh(asinh(Rational(3, 2))) == sqrt(Rational(13, 4))
def test_asinh_series():
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_rewrite():
assert asinh(x).rewrite(log) == log(x + sqrt(x**2 + 1))
def test_asinh():
assert asinh(+x) == +asinh(x)
assert asinh(-x) == -asinh(x)
assert asinh(-2) == -asinh(2)
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
assert asinh(zoo) == zoo
assert asinh(+I * (sqrt(3) - 1) / 2**Rational(3, 2)) == +pi * I / 12
assert asinh(-I * (sqrt(3) - 1) / 2**Rational(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
pytest.raises(ArgumentIndexError, lambda: asinh(x).fdiff(2))
def test_asinh():
assert asinh(x) == asinh(x)
assert asinh(-x) == -asinh(x)
assert asinh(-2) == -asinh(2)
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
assert asinh(zoo) == zoo
assert asinh(I*(sqrt(3) - 1)/2**Rational(3, 2)) == pi*I/12
assert asinh(-I*(sqrt(3) - 1)/2**Rational(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
pytest.raises(ArgumentIndexError, lambda: asinh(x).fdiff(2))
def test_sympyissue_4136():
assert cosh(asinh(Rational(3, 2))) == sqrt(Rational(13, 4))
def test_asinh_rewrite():
assert asinh(x).rewrite(log) == log(x + sqrt(x**2 + 1))
def test_asinh_series():
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_RR():
# Make sure the algorithm does the right thing if the ring is RR. See
# issue sympy/sympy#8685.
assert heurisch(sqrt(1 + 0.25*x**2), x, hints=[]) == \
0.5*x*sqrt(0.25*x**2 + 1) + 1.0*asinh(0.5*x)
def test_Function():
assert mathematica_code(f(x, y, z)) == 'f[x, y, z]'
assert mathematica_code(sin(x)**cos(x)) == 'Sin[x]^Cos[x]'
assert mathematica_code(sign(x)) == 'Sign[x]'
assert mathematica_code(atanh(x),
user_functions={'atanh':
'ArcTanh'}) == 'ArcTanh[x]'
assert (mathematica_code(meijerg(
((1, 1), (3, 4)), ((1, ), ()),
x)) == 'MeijerG[{{1, 1}, {3, 4}}, {{1}, {}}, x]')
assert (mathematica_code(hyper(
(1, 2, 3), (3, 4), x)) == 'HypergeometricPFQ[{1, 2, 3}, {3, 4}, x]')
assert mathematica_code(Min(x, y)) == 'Min[x, y]'
assert mathematica_code(Max(x, y)) == 'Max[x, y]'
assert mathematica_code(Max(x,
2)) == 'Max[2, x]' # issue sympy/sympy#15344
assert mathematica_code(binomial(x, y)) == 'Binomial[x, y]'
assert mathematica_code(log(x)) == 'Log[x]'
assert mathematica_code(tan(x)) == 'Tan[x]'
assert mathematica_code(cot(x)) == 'Cot[x]'
assert mathematica_code(asin(x)) == 'ArcSin[x]'
assert mathematica_code(acos(x)) == 'ArcCos[x]'
assert mathematica_code(atan(x)) == 'ArcTan[x]'
assert mathematica_code(acot(x)) == 'ArcCot[x]'
assert mathematica_code(sinh(x)) == 'Sinh[x]'
assert mathematica_code(cosh(x)) == 'Cosh[x]'
assert mathematica_code(tanh(x)) == 'Tanh[x]'
assert mathematica_code(coth(x)) == 'Coth[x]'
assert mathematica_code(asinh(x)) == 'ArcSinh[x]'
assert mathematica_code(acosh(x)) == 'ArcCosh[x]'
assert mathematica_code(atanh(x)) == 'ArcTanh[x]'
assert mathematica_code(acoth(x)) == 'ArcCoth[x]'
assert mathematica_code(sech(x)) == 'Sech[x]'
assert mathematica_code(csch(x)) == 'Csch[x]'
assert mathematica_code(erf(x)) == 'Erf[x]'
assert mathematica_code(erfi(x)) == 'Erfi[x]'
assert mathematica_code(erfc(x)) == 'Erfc[x]'
assert mathematica_code(conjugate(x)) == 'Conjugate[x]'
assert mathematica_code(re(x)) == 'Re[x]'
assert mathematica_code(im(x)) == 'Im[x]'
assert mathematica_code(polygamma(x, y)) == 'PolyGamma[x, y]'
assert mathematica_code(factorial(x)) == 'Factorial[x]'
assert mathematica_code(factorial2(x)) == 'Factorial2[x]'
assert mathematica_code(rf(x, y)) == 'Pochhammer[x, y]'
assert mathematica_code(gamma(x)) == 'Gamma[x]'
assert mathematica_code(zeta(x)) == 'Zeta[x]'
assert mathematica_code(Heaviside(x)) == 'UnitStep[x]'
assert mathematica_code(fibonacci(x)) == 'Fibonacci[x]'
assert mathematica_code(polylog(x, y)) == 'PolyLog[x, y]'
assert mathematica_code(loggamma(x)) == 'LogGamma[x]'
assert mathematica_code(uppergamma(x, y)) == 'Gamma[x, y]'
class MyFunc1(Function):
@classmethod
def eval(cls, x):
pass
class MyFunc2(Function):
@classmethod
def eval(cls, x, y):
pass
pytest.raises(
ValueError,
lambda: mathematica_code(MyFunc1(x),
user_functions={'MyFunc1': ['Myfunc1']}))
assert mathematica_code(MyFunc1(x),
user_functions={'MyFunc1':
'Myfunc1'}) == 'Myfunc1[x]'
assert mathematica_code(
MyFunc2(x, y),
user_functions={'MyFunc2':
[(lambda *x: False, 'Myfunc2')]}) == 'MyFunc2[x, y]'
