def test_acot():
assert acot(nan) == nan
assert acot.nargs == FiniteSet(1)
assert acot(-oo) == 0
assert acot(oo) == 0
assert acot(1) == pi/4
assert acot(0) == pi/2
assert acot(sqrt(3)/3) == pi/3
assert acot(1/sqrt(3)) == pi/3
assert acot(-1/sqrt(3)) == -pi/3
assert acot(x).diff(x) == -1/(1 + x**2)
assert acot(r).is_extended_real is True
assert acot(I*pi) == -I*acoth(pi)
assert acot(-2*I) == I*acoth(2)
assert acot(x).is_positive is None
assert acot(p).is_positive is True
assert acot(I).is_positive is False
assert acot(0, evaluate=False).is_rational is False
q = Symbol('q', rational=True)
assert acot(q).is_rational is False
assert acot(x).is_rational is None
pytest.raises(ArgumentIndexError, lambda: acot(x).fdiff(2))
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_acoth():
assert limit(acoth(x), x, 0) == -I*pi/2
assert limit(acoth(x), x, 0, 1) == I*pi/2
pytest.raises(PoleError, lambda: limit(acoth(x), x, 0, Reals))
assert limit(acoth(-x), x, 0) == I*pi/2
assert limit(acoth(x**2), x, 0, Reals) == -I*pi/2
assert limit(acoth(I*x), x, 0) == -I*pi/2
assert limit(acoth(I*x), x, 0, 1) == I*pi/2
pytest.raises(PoleError, lambda: limit(acoth(x - I*x), x, 0, Reals))
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_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_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_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():
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_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_acoth():
# TODO please write more tests -- see issue sympy/sympy#3751
# From http://functions.wolfram.com/ElementaryFunctions/ArcCoth/03/01/
# at specific points
# at specific points
assert acoth(0) == I * pi / 2
assert acoth(I) == -I * pi / 4
assert acoth(-I) == I * pi / 4
assert acoth(1) == oo
assert acoth(-1) == -oo
assert acoth(-2) == -acoth(2)
# at infinites
assert acoth(oo) == 0
assert acoth(-oo) == 0
assert acoth(I * oo) == 0
assert acoth(-I * oo) == 0
assert acoth(zoo) == 0
# properties
assert acoth(-x) == -acoth(x)
assert acoth(I / sqrt(3)) == -I * pi / 3
assert acoth(-I / sqrt(3)) == I * pi / 3
assert acoth(I * sqrt(3)) == -I * pi / 6
assert acoth(-I * sqrt(3)) == I * pi / 6
assert acoth(I * (1 + sqrt(2))) == -pi * I / 8
assert acoth(-I * (sqrt(2) + 1)) == pi * I / 8
assert acoth(I * (1 - sqrt(2))) == 3 * pi * I / 8
assert acoth(I * (sqrt(2) - 1)) == -3 * pi * I / 8
assert acoth(I * sqrt(5 + 2 * sqrt(5))) == -I * pi / 10
assert acoth(-I * sqrt(5 + 2 * sqrt(5))) == I * pi / 10
assert acoth(I * (2 + sqrt(3))) == -pi * I / 12
assert acoth(-I * (2 + sqrt(3))) == pi * I / 12
assert acoth(I * (2 - sqrt(3))) == -5 * pi * I / 12
assert acoth(I * (sqrt(3) - 2)) == 5 * pi * I / 12
pytest.raises(ArgumentIndexError, lambda: acoth(x).fdiff(2))
def test_acoth():
# TODO please write more tests -- see issue sympy/sympy#3751
# From http://functions.wolfram.com/ElementaryFunctions/ArcCoth/03/01/
# at specific points
# at specific points
assert acoth(0) == I*pi/2
assert acoth(I) == -I*pi/4
assert acoth(-I) == I*pi/4
assert acoth(1) == oo
assert acoth(-1) == -oo
assert acoth(-2) == -acoth(2)
# at infinites
assert acoth(oo) == 0
assert acoth(-oo) == 0
assert acoth(I*oo) == 0
assert acoth(-I*oo) == 0
assert acoth(zoo) == 0
# properties
assert acoth(-x) == -acoth(x)
assert acoth(I/sqrt(3)) == -I*pi/3
assert acoth(-I/sqrt(3)) == I*pi/3
assert acoth(I*sqrt(3)) == -I*pi/6
assert acoth(-I*sqrt(3)) == I*pi/6
assert acoth(I*(1 + sqrt(2))) == -pi*I/8
assert acoth(-I*(sqrt(2) + 1)) == pi*I/8
assert acoth(I*(1 - sqrt(2))) == 3*pi*I/8
assert acoth(I*(sqrt(2) - 1)) == -3*pi*I/8
assert acoth(I*sqrt(5 + 2*sqrt(5))) == -I*pi/10
assert acoth(-I*sqrt(5 + 2*sqrt(5))) == I*pi/10
assert acoth(I*(2 + sqrt(3))) == -pi*I/12
assert acoth(-I*(2 + sqrt(3))) == pi*I/12
assert acoth(I*(2 - sqrt(3))) == -5*pi*I/12
assert acoth(I*(sqrt(3) - 2)) == 5*pi*I/12
pytest.raises(ArgumentIndexError, lambda: acoth(x).fdiff(2))
def test_acoth_series():
# see sympy/sympy#3663
assert (acoth(x).series(x, n=10) == -I * pi / 2 + x + x**3 / 3 + x**5 / 5 +
x**7 / 7 + x**9 / 9 + O(x**10))
def test_acoth_rewrite():
assert acoth(x).rewrite(log) == (log((x + 1) / x) - log((x - 1) / x)) / 2
def test_acoth_series():
x = Symbol('x')
assert acoth(x).series(x, 0, 10) == \
I*pi/2 + x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
def test_acoth_series():
assert acoth(x).series(x, 0, 10) == \
I*pi/2 + x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
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_acoth():
# TODO please write more tests -- see issue 3751
# From http://functions.wolfram.com/ElementaryFunctions/ArcCoth/03/01/
# at specific points
x = Symbol('x')
# at specific points
assert acoth(0) == I * pi / 2
assert acoth(I) == -I * pi / 4
assert acoth(-I) == I * pi / 4
assert acoth(1) == oo
assert acoth(-1) == -oo
# at infinites
assert acoth(oo) == 0
assert acoth(-oo) == 0
assert acoth(I * oo) == 0
assert acoth(-I * oo) == 0
assert acoth(zoo) == 0
# properties
assert acoth(-x) == -acoth(x)
assert acoth(I / sqrt(3)) == -I * pi / 3
assert acoth(-I / sqrt(3)) == I * pi / 3
assert acoth(I * sqrt(3)) == -I * pi / 6
assert acoth(-I * sqrt(3)) == I * pi / 6
assert acoth(I * (1 + sqrt(2))) == -pi * I / 8
assert acoth(-I * (sqrt(2) + 1)) == pi * I / 8
assert acoth(I * (1 - sqrt(2))) == 3 * pi * I / 8
assert acoth(I * (sqrt(2) - 1)) == -3 * pi * I / 8
assert acoth(I * sqrt(5 + 2 * sqrt(5))) == -I * pi / 10
assert acoth(-I * sqrt(5 + 2 * sqrt(5))) == I * pi / 10
assert acoth(I * (2 + sqrt(3))) == -pi * I / 12
assert acoth(-I * (2 + sqrt(3))) == pi * I / 12
assert acoth(I * (2 - sqrt(3))) == -5 * pi * I / 12
assert acoth(I * (sqrt(3) - 2)) == 5 * pi * I / 12
