def test_issue_75():
assert limit(abs(log(x)), x, oo) == oo
assert limit(tan(abs(pi/2 + 1/x))/acosh(pi/2 + 1/x), x, oo) == -oo
assert limit(tan(abs(pi/2 - 1/x))/acosh(pi/2 - 1/x), x, oo) == +oo
assert limit(abs(log(2 + 1/x)) - log(2 + 1/x), x, oo) == 0
assert limit(abs(log(2 - 1/x)) - log(2 - 1/x), x, oo) == 0
def test_diofantissue_75():
assert gruntz(abs(log(x)), x) == oo
assert gruntz(tan(abs(pi / 2 + 1 / x)) / acosh(pi / 2 + 1 / x), x) == -oo
assert gruntz(tan(abs(pi / 2 - 1 / x)) / acosh(pi / 2 - 1 / x), x) == +oo
assert gruntz(abs(log(2 + 1 / x)) - log(2 + 1 / x), x) == 0
assert gruntz(abs(log(2 - 1 / x)) - log(2 - 1 / x), x) == 0
def test_acosh_series():
x = Symbol('x')
assert acosh(x).series(x, 0, 8) == \
-I*x + pi*I/2 - I*x**3/6 - 3*I*x**5/40 - 5*I*x**7/112 + O(x**8)
t5 = acosh(x).taylor_term(5, x)
assert t5 == -3 * I * x**5 / 40
assert acosh(x).taylor_term(7, x, t5, 0) == -5 * I * x**7 / 112
def test_diofantissue_75():
assert gruntz(abs(log(x)), x) == oo
assert gruntz(tan(abs(pi/2 + 1/x))/acosh(pi/2 + 1/x), x) == -oo
assert gruntz(tan(abs(pi/2 - 1/x))/acosh(pi/2 - 1/x), x) == +oo
assert gruntz(abs(log(2 + 1/x)) - log(2 + 1/x), x) == 0
assert gruntz(abs(log(2 - 1/x)) - log(2 - 1/x), x) == 0
def test_acosh_series():
assert (acosh(x).series(x, 0, 8) == -I * x + pi * I / 2 - I * x**3 / 6 -
3 * I * x**5 / 40 - 5 * I * x**7 / 112 + O(x**8))
t5 = acosh(x).taylor_term(5, x)
assert t5 == -3 * I * x**5 / 40
assert acosh(x).taylor_term(7, x, t5, 0) == -5 * I * x**7 / 112
assert (acosh(x).series(x, x0=1, n=2) == sqrt(2) * sqrt(x - 1) -
sqrt(2) * sqrt(x - 1)**3 / 12 + O((x - 1)**2, (x, 1)))
def test_acosh():
assert limit(acosh(I*x), x, 0) == 2*log(1 + I) - log(2)
assert limit(acosh(I*x), x, 0, 1) == 2*log(1 - I) - log(2)
assert limit(acosh(-I*x), x, 0) == 2*log(1 - I) - log(2)
assert limit(acosh(-I*x), x, 0, 1) == 2*log(1 + I) - log(2)
pytest.raises(PoleError, lambda: limit(acosh(I*x), x, 0, Reals))
assert limit(acosh(x - I*x), x, 0) == 2*log(1 - I) - log(2)
assert limit(acosh(x - I*x), x, 0, 1) == 2*log(1 + I) - log(2)
pytest.raises(PoleError, lambda: limit(acosh(x - I*x), x, 0, Reals))
assert limit(acosh(I*x**2), x, 0, Reals) == 2*log(1 + I) - log(2)
assert limit(acosh(x**2 - I*x**2), x, 0, Reals) == 2*log(1 - I) - log(2)
def test_intractable():
assert limit(1/gamma(x), x, oo) == 0
assert limit(1/loggamma(x), x, oo) == 0
assert limit(gamma(x)/loggamma(x), x, oo) == oo
assert limit(exp(gamma(x))/gamma(x), x, oo) == oo
assert limit(gamma(3 + 1/x), x, oo) == 2
assert limit(gamma(Rational(1, 7) + 1/x), x, oo) == gamma(Rational(1, 7))
assert limit(log(x**x)/log(gamma(x)), x, oo) == 1
assert limit(log(gamma(gamma(x)))/exp(x), x, oo) == oo
assert limit(acosh(1 + 1/x)*sqrt(x), x, oo) == sqrt(2)
# issue sympy/sympy#10804
assert limit(2*airyai(x)*root(x, 4) *
exp(2*x**Rational(3, 2)/3), x, oo) == 1/sqrt(pi)
assert limit(airybi(x)*root(x, 4) *
exp(-2*x**Rational(3, 2)/3), x, oo) == 1/sqrt(pi)
assert limit(airyai(1/x), x, oo) == (3**Rational(5, 6) *
gamma(Rational(1, 3))/(6*pi))
assert limit(airybi(1/x), x, oo) == cbrt(3)*gamma(Rational(1, 3))/(2*pi)
assert limit(airyai(2 + 1/x), x, oo) == airyai(2)
assert limit(airybi(2 + 1/x), x, oo) == airybi(2)
# issue sympy/sympy#10976
assert limit(erf(m/x)/erf(1/x), x, oo) == m
assert limit(Max(x**2, x, exp(x))/x, x, oo) == oo
def test_messy():
assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi/2)/s, 0, True)
assert laplace_transform(Shi(x), x, s) == (acoth(s)/s, 1, True)
# where should the logs be simplified?
assert laplace_transform(Chi(x), x, s) == \
((log(s**(-2)) - log((s**2 - 1)/s**2))/(2*s), 1, True)
# TODO maybe simplify the inequalities?
assert laplace_transform(besselj(a, x), x, s)[1:] == \
(0, And(Integer(0) < re(a/2) + Rational(1, 2), Integer(0) < re(a/2) + 1))
# NOTE s < 0 can be done, but argument reduction is not good enough yet
assert fourier_transform(besselj(1, x)/x, x, s, noconds=False) == \
(Piecewise((0, 4*abs(pi**2*s**2) > 1),
(2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
# TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
# - folding could be better
assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
log(1 + sqrt(2))
assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
log(Rational(1, 2) + sqrt(2)/2)
assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
Piecewise((-acosh(1/x), 1 < abs(x**(-2))), (I*asin(1/x), True))
def test_messy():
assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi / 2) / s, 0, True)
assert laplace_transform(Shi(x), x, s) == (acoth(s) / s, 1, True)
# where should the logs be simplified?
assert laplace_transform(Chi(x), x, s) == \
((log(s**(-2)) - log((s**2 - 1)/s**2))/(2*s), 1, True)
# TODO maybe simplify the inequalities?
assert laplace_transform(besselj(a, x), x, s)[1:] == \
(0, And(Integer(0) < re(a/2) + Rational(1, 2), Integer(0) < re(a/2) + 1))
# NOTE s < 0 can be done, but argument reduction is not good enough yet
assert fourier_transform(besselj(1, x)/x, x, s, noconds=False) == \
(Piecewise((0, 4*abs(pi**2*s**2) > 1),
(2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
# TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
# - folding could be better
assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
log(1 + sqrt(2))
assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
log(Rational(1, 2) + sqrt(2)/2)
assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
Piecewise((-acosh(1/x), 1 < abs(x**(-2))), (I*asin(1/x), True))
def test_numpy_numexpr():
a, b, c = numpy.random.randn(3, 128, 128)
# ensure that numpy and numexpr return same value for complicated expression
expr = (sin(x) + cos(y) + tan(z)**2 + Abs(z - y)*acos(sin(y*z)) +
Abs(y - z)*acosh(2 + exp(y - x)) - sqrt(x**2 + I*y**2))
npfunc = lambdify((x, y, z), expr, modules='numpy')
nefunc = lambdify((x, y, z), expr, modules='numexpr')
assert numpy.allclose(npfunc(a, b, c), nefunc(a, b, c))
def test_numpy_numexpr():
a, b, c = numpy.random.randn(3, 128, 128)
# ensure that numpy and numexpr return same value for complicated expression
expr = (sin(x) + cos(y) + tan(z)**2 + Abs(z - y)*acos(sin(y*z)) +
Abs(y - z)*acosh(2 + exp(y - x)) - sqrt(x**2 + I*y**2))
npfunc = lambdify((x, y, z), expr, modules='numpy')
nefunc = lambdify((x, y, z), expr, modules='numexpr')
assert numpy.allclose(npfunc(a, b, c), nefunc(a, b, c))
def test_sympyissue_4492():
assert simplify(integrate(x**2 * sqrt(5 - x**2), x)) == Piecewise(
(I * (2 * x**5 - 15 * x**3 + 25 * x -
25 * sqrt(x**2 - 5) * acosh(sqrt(5) * x / 5)) /
(8 * sqrt(x**2 - 5)), 1 < Abs(x**2) / 5),
((-2 * x**5 + 15 * x**3 - 25 * x +
25 * sqrt(-x**2 + 5) * asin(sqrt(5) * x / 5)) /
(8 * sqrt(-x**2 + 5)), True))
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_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).series(x, n=7) == x + x**3/6 + x**5/120 + O(x**7)
assert cosh(x).series(x) == 1 + x**2/2 + x**4/24 + O(x**6)
assert tanh(x).series(x, n=7) == x - x**3/3 + 2*x**5/15 + O(x**7)
assert coth(x).series(x, n=7) == \
1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**7)
assert asinh(x).series(x, n=7) == x - x**3/6 + 3*x**5/40 + O(x**7)
assert acosh(x).series(x, n=7) == \
pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**7)
assert atanh(x).series(x, n=7) == x + x**3/3 + x**5/5 + O(x**7)
assert acoth(x).series(x, n=7) == -I*pi/2 + x + x**3/3 + x**5/5 + O(x**7)
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_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_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_leading_term():
assert cosh(x).as_leading_term(x) == 1
assert coth(x).as_leading_term(x) == 1 / x
assert acosh(x).as_leading_term(x) == I * pi / 2
assert acoth(x).as_leading_term(x) == -I * pi / 2
for func in [sinh, tanh, asinh, atanh]:
assert func(x).as_leading_term(x) == x
for func in [sinh, cosh, tanh, coth, asinh, acosh, atanh, acoth]:
for arg in (1 / x, Rational(1, 2)):
eq = func(arg)
assert eq.as_leading_term(x) == eq
for func in [csch, sech]:
eq = func(Rational(1, 2))
assert eq.as_leading_term(x) == eq
assert csch(x).as_leading_term(x) == 1 / x
def test_leading_term():
assert cosh(x).as_leading_term(x) == 1
assert coth(x).as_leading_term(x) == 1/x
assert acosh(x).as_leading_term(x) == I*pi/2
assert acoth(x).as_leading_term(x) == I*pi/2
for func in [sinh, tanh, asinh, atanh]:
assert func(x).as_leading_term(x) == x
for func in [sinh, cosh, tanh, coth, asinh, acosh, atanh, acoth]:
for arg in (1/x, Rational(1, 2)):
eq = func(arg)
assert eq.as_leading_term(x) == eq
for func in [csch, sech]:
eq = func(Rational(1, 2))
assert eq.as_leading_term(x) == eq
assert csch(x).as_leading_term(x) == 1/x
def test_leading_term():
x = Symbol('x')
assert cosh(x).as_leading_term(x) == 1
assert coth(x).as_leading_term(x) == 1 / x
assert acosh(x).as_leading_term(x) == I * pi / 2
assert acoth(x).as_leading_term(x) == I * pi / 2
for func in [sinh, tanh, asinh, atanh]:
assert func(x).as_leading_term(x) == x
for func in [sinh, cosh, tanh, coth, asinh, acosh, atanh, acoth]:
for arg in (1 / x, S.Half):
eq = func(arg)
assert eq.as_leading_term(x) == eq
for func in [csch, sech]:
eq = func(S.Half)
assert eq.as_leading_term(x) == eq
def test_intractable():
assert gruntz(1/gamma(x), x) == 0
assert gruntz(1/loggamma(x), x) == 0
assert gruntz(gamma(x)/loggamma(x), x) == oo
assert gruntz(exp(gamma(x))/gamma(x), x) == oo
assert gruntz(gamma(3 + 1/x), x) == 2
assert gruntz(gamma(Rational(1, 7) + 1/x), x) == gamma(Rational(1, 7))
assert gruntz(log(x**x)/log(gamma(x)), x) == 1
assert gruntz(log(gamma(gamma(x)))/exp(x), x) == oo
assert gruntz(acosh(1 + 1/x)*sqrt(x), x) == sqrt(2)
# issue sympy/sympy#10804
assert gruntz(2*airyai(x)*root(x, 4) *
exp(2*x**Rational(3, 2)/3), x) == 1/sqrt(pi)
assert gruntz(airybi(x)*root(x, 4) *
exp(-2*x**Rational(3, 2)/3), x) == 1/sqrt(pi)
assert gruntz(airyai(1/x), x) == (3**Rational(5, 6) *
gamma(Rational(1, 3))/(6*pi))
assert gruntz(airybi(1/x), x) == cbrt(3)*gamma(Rational(1, 3))/(2*pi)
assert gruntz(airyai(2 + 1/x), x) == airyai(2)
assert gruntz(airybi(2 + 1/x), x) == airybi(2)
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_acosh():
# TODO please write more tests -- see issue sympy/sympy#3751
# From http://functions.wolfram.com/ElementaryFunctions/ArcCosh/03/01/
# at specific points
assert acosh(-x) == acosh(-x)
assert acosh(1) == 0
assert acosh(-1) == pi * I
assert acosh(0) == I * pi / 2
assert acosh(Rational(1, 2)) == I * pi / 3
assert acosh(Rational(-1, 2)) == 2 * pi * I / 3
assert acosh(+oo) == oo
assert acosh(-oo) == oo
assert acosh(+I * oo) == oo
assert acosh(-I * oo) == oo
assert acosh(zoo) == oo
assert acosh(I) == log(I * (1 + sqrt(2)))
assert acosh(-I) == log(-I * (1 + sqrt(2)))
assert acosh((sqrt(3) - 1) / (2 * sqrt(2))) == 5 * pi * I / 12
assert acosh(-(sqrt(3) - 1) / (2 * sqrt(2))) == 7 * pi * I / 12
assert acosh(sqrt(2) / 2) == I * pi / 4
assert acosh(-sqrt(2) / 2) == 3 * I * pi / 4
assert acosh(sqrt(3) / 2) == I * pi / 6
assert acosh(-sqrt(3) / 2) == 5 * I * pi / 6
assert acosh(sqrt(2 + sqrt(2)) / 2) == I * pi / 8
assert acosh(-sqrt(2 + sqrt(2)) / 2) == 7 * I * pi / 8
assert acosh(sqrt(2 - sqrt(2)) / 2) == 3 * I * pi / 8
assert acosh(-sqrt(2 - sqrt(2)) / 2) == 5 * I * pi / 8
assert acosh((1 + sqrt(3)) / (2 * sqrt(2))) == I * pi / 12
assert acosh(-(1 + sqrt(3)) / (2 * sqrt(2))) == 11 * I * pi / 12
assert acosh((sqrt(5) + 1) / 4) == I * pi / 5
assert acosh(-(sqrt(5) + 1) / 4) == 4 * I * pi / 5
assert str(acosh(5 * I).evalf(6)) == '2.31244 + 1.5708*I'
assert str(acosh(-5 * I).evalf(6)) == '2.31244 - 1.5708*I'
pytest.raises(ArgumentIndexError, lambda: acosh(x).fdiff(2))
def test_acosh_series():
assert acosh(x).series(x, 0, 8) == \
-I*x + pi*I/2 - I*x**3/6 - 3*I*x**5/40 - 5*I*x**7/112 + O(x**8)
t5 = acosh(x).taylor_term(5, x)
assert t5 == - 3*I*x**5/40
assert acosh(x).taylor_term(7, x, t5, 0) == - 5*I*x**7/112
def test_sympyissue_22986():
assert limit(acosh(1 + 1/x)*sqrt(x), x, oo) == sqrt(2)
def test_sympyissue_4492():
assert simplify(integrate(x**2 * sqrt(5 - x**2), x)) == Piecewise(
(I*(2*x**5 - 15*x**3 + 25*x - 25*sqrt(x**2 - 5)*acosh(sqrt(5)*x/5)) /
(8*sqrt(x**2 - 5)), 1 < Abs(x**2)/5),
((-2*x**5 + 15*x**3 - 25*x + 25*sqrt(-x**2 + 5)*asin(sqrt(5)*x/5)) /
(8*sqrt(-x**2 + 5)), True))
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]'
def test_sympyissue_4551():
assert (integrate(1/(x*sqrt(1 - x**2)), x) ==
Piecewise((-acosh(1/x), abs(x**-2) > 1), (I*asin(1/x), True)))
def test_acosh_infinities():
assert acosh(oo) == oo
assert acosh(-oo) == oo
assert acosh(I * oo) == oo
assert acosh(-I * oo) == oo
def test_acosh_rewrite():
assert acosh(x).rewrite(log) == 2 * log(sqrt(x - 1) + sqrt(x + 1)) - log(2)
def test_sympyissue_4551():
assert (integrate(1/(x*sqrt(1 - x**2)), x) ==
Piecewise((-acosh(1/x), Abs(x**(-2)) > 1), (I*asin(1/x), True)))
def test_acosh():
# TODO please write more tests -- see issue 3751
# From http://functions.wolfram.com/ElementaryFunctions/ArcCosh/03/01/
# at specific points
x = Symbol('x')
assert acosh(-x) == acosh(-x)
assert acosh(1) == 0
assert acosh(-1) == pi * I
assert acosh(0) == I * pi / 2
assert acosh(Rational(1, 2)) == I * pi / 3
assert acosh(Rational(-1, 2)) == 2 * pi * I / 3
assert acosh(zoo) == oo
assert acosh(I) == log(I * (1 + sqrt(2)))
assert acosh(-I) == log(-I * (1 + sqrt(2)))
assert acosh((sqrt(3) - 1) / (2 * sqrt(2))) == 5 * pi * I / 12
assert acosh(-(sqrt(3) - 1) / (2 * sqrt(2))) == 7 * pi * I / 12
assert acosh(sqrt(2) / 2) == I * pi / 4
assert acosh(-sqrt(2) / 2) == 3 * I * pi / 4
assert acosh(sqrt(3) / 2) == I * pi / 6
assert acosh(-sqrt(3) / 2) == 5 * I * pi / 6
assert acosh(sqrt(2 + sqrt(2)) / 2) == I * pi / 8
assert acosh(-sqrt(2 + sqrt(2)) / 2) == 7 * I * pi / 8
assert acosh(sqrt(2 - sqrt(2)) / 2) == 3 * I * pi / 8
assert acosh(-sqrt(2 - sqrt(2)) / 2) == 5 * I * pi / 8
assert acosh((1 + sqrt(3)) / (2 * sqrt(2))) == I * pi / 12
assert acosh(-(1 + sqrt(3)) / (2 * sqrt(2))) == 11 * I * pi / 12
assert acosh((sqrt(5) + 1) / 4) == I * pi / 5
assert acosh(-(sqrt(5) + 1) / 4) == 4 * I * pi / 5
assert str(acosh(5 * I).n(6)) == '2.31244 + 1.5708*I'
assert str(acosh(-5 * I).n(6)) == '2.31244 - 1.5708*I'
def test_acosh():
# TODO please write more tests -- see issue sympy/sympy#3751
# From http://functions.wolfram.com/ElementaryFunctions/ArcCosh/03/01/
# at specific points
assert acosh(-x) == acosh(-x)
assert acosh(1) == 0
assert acosh(-1) == pi*I
assert acosh(0) == I*pi/2
assert acosh(Rational(1, 2)) == I*pi/3
assert acosh(Rational(-1, 2)) == 2*pi*I/3
assert acosh(+oo) == oo
assert acosh(-oo) == oo
assert acosh(+I*oo) == oo
assert acosh(-I*oo) == oo
assert acosh(zoo) == oo
assert acosh(I) == log(I*(1 + sqrt(2)))
assert acosh(-I) == log(-I*(1 + sqrt(2)))
assert acosh((sqrt(3) - 1)/(2*sqrt(2))) == 5*pi*I/12
assert acosh(-(sqrt(3) - 1)/(2*sqrt(2))) == 7*pi*I/12
assert acosh(sqrt(2)/2) == I*pi/4
assert acosh(-sqrt(2)/2) == 3*I*pi/4
assert acosh(sqrt(3)/2) == I*pi/6
assert acosh(-sqrt(3)/2) == 5*I*pi/6
assert acosh(sqrt(2 + sqrt(2))/2) == I*pi/8
assert acosh(-sqrt(2 + sqrt(2))/2) == 7*I*pi/8
assert acosh(sqrt(2 - sqrt(2))/2) == 3*I*pi/8
assert acosh(-sqrt(2 - sqrt(2))/2) == 5*I*pi/8
assert acosh((1 + sqrt(3))/(2*sqrt(2))) == I*pi/12
assert acosh(-(1 + sqrt(3))/(2*sqrt(2))) == 11*I*pi/12
assert acosh((sqrt(5) + 1)/4) == I*pi/5
assert acosh(-(sqrt(5) + 1)/4) == 4*I*pi/5
assert str(acosh(5*I).evalf(6)) == '2.31244 + 1.5708*I'
assert str(acosh(-5*I).evalf(6)) == '2.31244 - 1.5708*I'
pytest.raises(ArgumentIndexError, lambda: acosh(x).fdiff(2))
