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
assert asech(x).inverse() == sech
assert acsch(x).inverse() == csch
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_asech_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: asech(x).fdiff(2))
def test_asech_rewrite():
x = Symbol('x')
assert asech(x).rewrite(log) == log(1 / x +
sqrt(1 / x - 1) * sqrt(1 / x + 1))
def test_asech_series():
x = Symbol('x')
t6 = asech(x).expansion_term(6, x)
assert t6 == -5 * x**6 / 96
assert asech(x).expansion_term(8, x, t6, 0) == -35 * x**8 / 1024
def test_asech():
x = Symbol('x')
assert unchanged(asech, -x)
# values at fixed points
assert asech(1) == 0
assert asech(-1) == pi * I
assert asech(0) is oo
assert asech(2) == I * pi / 3
assert asech(-2) == 2 * I * pi / 3
assert asech(nan) is nan
# at infinites
assert asech(oo) == I * pi / 2
assert asech(-oo) == I * pi / 2
assert asech(zoo) == I * AccumBounds(-pi / 2, pi / 2)
assert asech(I) == log(1 + sqrt(2)) - I * pi / 2
assert asech(-I) == log(1 + sqrt(2)) + I * pi / 2
assert asech(sqrt(2) - sqrt(6)) == 11 * I * pi / 12
assert asech(sqrt(2 - 2 / sqrt(5))) == I * pi / 10
assert asech(-sqrt(2 - 2 / sqrt(5))) == 9 * I * pi / 10
assert asech(2 / sqrt(2 + sqrt(2))) == I * pi / 8
assert asech(-2 / sqrt(2 + sqrt(2))) == 7 * I * pi / 8
assert asech(sqrt(5) - 1) == I * pi / 5
assert asech(1 - sqrt(5)) == 4 * I * pi / 5
assert asech(-sqrt(2 * (2 + sqrt(2)))) == 5 * I * pi / 8
# properties
# asech(x) == acosh(1/x)
assert asech(sqrt(2)) == acosh(1 / sqrt(2))
assert asech(2 / sqrt(3)) == acosh(sqrt(3) / 2)
assert asech(2 / sqrt(2 + sqrt(2))) == acosh(sqrt(2 + sqrt(2)) / 2)
assert asech(2) == acosh(S.Half)
# asech(x) == I*acos(1/x)
# (Note: the exact formula is asech(x) == +/- I*acos(1/x))
assert asech(-sqrt(2)) == I * acos(-1 / sqrt(2))
assert asech(-2 / sqrt(3)) == I * acos(-sqrt(3) / 2)
assert asech(-S(2)) == I * acos(Rational(-1, 2))
assert asech(-2 / sqrt(2)) == I * acos(-sqrt(2) / 2)
# sech(asech(x)) / x == 1
assert expand_mul(sech(asech(sqrt(6) - sqrt(2))) /
(sqrt(6) - sqrt(2))) == 1
assert expand_mul(sech(asech(sqrt(6) + sqrt(2))) /
(sqrt(6) + sqrt(2))) == 1
assert (sech(asech(sqrt(2 + 2 / sqrt(5)))) /
(sqrt(2 + 2 / sqrt(5)))).simplify() == 1
assert (sech(asech(-sqrt(2 + 2 / sqrt(5)))) /
(-sqrt(2 + 2 / sqrt(5)))).simplify() == 1
assert (sech(asech(sqrt(2 * (2 + sqrt(2))))) /
(sqrt(2 * (2 + sqrt(2))))).simplify() == 1
assert expand_mul(sech(asech(1 + sqrt(5))) / (1 + sqrt(5))) == 1
assert expand_mul(sech(asech(-1 - sqrt(5))) / (-1 - sqrt(5))) == 1
assert expand_mul(sech(asech(-sqrt(6) - sqrt(2))) /
(-sqrt(6) - sqrt(2))) == 1
# numerical evaluation
assert str(asech(5 * I).n(6)) == '0.19869 - 1.5708*I'
assert str(asech(-5 * I).n(6)) == '0.19869 + 1.5708*I'
def test_fps__logarithmic_singularity_fail():
f = asech(x) # Algorithms for computing limits probably needs improvemnts
assert fps(f, x) == log(2) - log(x) - x**2/4 - 3*x**4/64 + O(x**6)