def test_hyper_as_trig():
from diofant.simplify.fu import _osborne as o, _osbornei as i, TR12
eq = sinh(x)**2 + cosh(x)**2
t, f = hyper_as_trig(eq)
assert f(fu(t)) == cosh(2*x)
e, f = hyper_as_trig(tanh(x + y))
assert f(TR12(e)) == (tanh(x) + tanh(y))/(tanh(x)*tanh(y) + 1)
d = Dummy()
assert o(sinh(x), d) == I*sin(x*d)
assert o(tanh(x), d) == I*tan(x*d)
assert o(coth(x), d) == cot(x*d)/I
assert o(cosh(x), d) == cos(x*d)
for func in (sinh, cosh, tanh, coth):
h = func(pi)
assert i(o(h, d), d) == h
# /!\ the _osborne functions are not meant to work
# in the o(i(trig, d), d) direction so we just check
# that they work as they are supposed to work
assert i(cos(x*y), y) == cosh(x)
assert i(sin(x*y), y) == sinh(x)/I
assert i(tan(x*y), y) == tanh(x)/I
assert i(cot(x*y), y) == coth(x)*I
assert i(sec(x*y), y) == 1/cosh(x)
assert i(csc(x*y), y) == I/sinh(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_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_tan_rewrite():
neg_exp, pos_exp = exp(-x*I), exp(x*I)
assert tan(x).rewrite(exp) == I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x)
assert tan(x).rewrite(cos) == -cos(x + pi/2)/cos(x)
assert tan(x).rewrite(cot) == 1/cot(x)
assert (tan(sinh(x)).rewrite(exp).subs({x: 3}).evalf() ==
tan(x).rewrite(exp).subs({x: sinh(3)}).evalf())
assert (tan(cosh(x)).rewrite(exp).subs({x: 3}).evalf() ==
tan(x).rewrite(exp).subs({x: cosh(3)}).evalf())
assert (tan(tanh(x)).rewrite(exp).subs({x: 3}).evalf() ==
tan(x).rewrite(exp).subs({x: tanh(3)}).evalf())
assert (tan(coth(x)).rewrite(exp).subs({x: 3}).evalf() ==
tan(x).rewrite(exp).subs({x: coth(3)}).evalf())
assert (tan(sin(x)).rewrite(exp).subs({x: 3}).evalf() ==
tan(x).rewrite(exp).subs({x: sin(3)}).evalf())
assert (tan(cos(x)).rewrite(exp).subs({x: 3}).evalf() ==
tan(x).rewrite(exp).subs({x: cos(3)}).evalf())
assert (tan(tan(x)).rewrite(exp).subs({x: 3}).evalf() ==
tan(x).rewrite(exp).subs({x: tan(3)}).evalf())
assert (tan(cot(x)).rewrite(exp).subs({x: 3}).evalf() ==
tan(x).rewrite(exp).subs({x: cot(3)}).evalf())
assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
assert tan(x).rewrite(Pow) == tan(x)
assert 0 == (cos(pi/34)*tan(pi/34) - sin(pi/34)).rewrite(sqrt)
assert 0 == (cos(pi/17)*tan(pi/17) - sin(pi/17)).rewrite(sqrt)
assert tan(pi/19).rewrite(sqrt) == tan(pi/19)
assert tan(8*pi/19).rewrite(sqrt) == tan(8*pi/19)
def test_coth_rewrite():
x = Symbol('x')
assert coth(x).rewrite(exp) == (exp(x) + exp(-x))/(exp(x) - exp(-x)) \
== coth(x).rewrite('tractable')
assert coth(x).rewrite(sinh) == -I * sinh(I * pi / 2 - x) / sinh(x)
assert coth(x).rewrite(cosh) == -I * cosh(x) / cosh(I * pi / 2 - x)
assert coth(x).rewrite(tanh) == 1 / tanh(x)
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_gruntz_hyperbolic():
assert gruntz(cosh(x), x) == oo
assert gruntz(cosh(-x), x) == oo
assert gruntz(sinh(x), x) == oo
assert gruntz(sinh(-x), x) == -oo
assert gruntz(2 * cosh(x) * exp(x), x) == oo
assert gruntz(2 * cosh(-x) * exp(-x), x) == 1
assert gruntz(2 * sinh(x) * exp(x), x) == oo
assert gruntz(2 * sinh(-x) * exp(-x), x) == -1
assert gruntz(tanh(x), x) == 1
assert gruntz(tanh(-x), x) == -1
assert gruntz(coth(x), x) == 1
assert gruntz(coth(-x), x) == -1
def test_gruntz_hyperbolic():
assert gruntz(cosh(x), x) == oo
assert gruntz(cosh(-x), x) == oo
assert gruntz(sinh(x), x) == oo
assert gruntz(sinh(-x), x) == -oo
assert gruntz(2*cosh(x)*exp(x), x) == oo
assert gruntz(2*cosh(-x)*exp(-x), x) == 1
assert gruntz(2*sinh(x)*exp(x), x) == oo
assert gruntz(2*sinh(-x)*exp(-x), x) == -1
assert gruntz(tanh(x), x) == 1
assert gruntz(tanh(-x), x) == -1
assert gruntz(coth(x), x) == 1
assert gruntz(coth(-x), x) == -1
def test_trigsimp1a():
assert trigsimp(sin(2)**2*cos(3)*exp(2)/cos(2)**2) == tan(2)**2*cos(3)*exp(2)
assert trigsimp(tan(2)**2*cos(3)*exp(2)*cos(2)**2) == sin(2)**2*cos(3)*exp(2)
assert trigsimp(cot(2)*cos(3)*exp(2)*sin(2)) == cos(3)*exp(2)*cos(2)
assert trigsimp(tan(2)*cos(3)*exp(2)/sin(2)) == cos(3)*exp(2)/cos(2)
assert trigsimp(cot(2)*cos(3)*exp(2)/cos(2)) == cos(3)*exp(2)/sin(2)
assert trigsimp(cot(2)*cos(3)*exp(2)*tan(2)) == cos(3)*exp(2)
assert trigsimp(sinh(2)*cos(3)*exp(2)/cosh(2)) == tanh(2)*cos(3)*exp(2)
assert trigsimp(tanh(2)*cos(3)*exp(2)*cosh(2)) == sinh(2)*cos(3)*exp(2)
assert trigsimp(coth(2)*cos(3)*exp(2)*sinh(2)) == cosh(2)*cos(3)*exp(2)
assert trigsimp(tanh(2)*cos(3)*exp(2)/sinh(2)) == cos(3)*exp(2)/cosh(2)
assert trigsimp(coth(2)*cos(3)*exp(2)/cosh(2)) == cos(3)*exp(2)/sinh(2)
assert trigsimp(coth(2)*cos(3)*exp(2)*tanh(2)) == cos(3)*exp(2)
def test_trigsimp1a():
assert trigsimp(sin(2)**2*cos(3)*exp(2)/cos(2)**2) == tan(2)**2*cos(3)*exp(2)
assert trigsimp(tan(2)**2*cos(3)*exp(2)*cos(2)**2) == sin(2)**2*cos(3)*exp(2)
assert trigsimp(cot(2)*cos(3)*exp(2)*sin(2)) == cos(3)*exp(2)*cos(2)
assert trigsimp(tan(2)*cos(3)*exp(2)/sin(2)) == cos(3)*exp(2)/cos(2)
assert trigsimp(cot(2)*cos(3)*exp(2)/cos(2)) == cos(3)*exp(2)/sin(2)
assert trigsimp(cot(2)*cos(3)*exp(2)*tan(2)) == cos(3)*exp(2)
assert trigsimp(sinh(2)*cos(3)*exp(2)/cosh(2)) == tanh(2)*cos(3)*exp(2)
assert trigsimp(tanh(2)*cos(3)*exp(2)*cosh(2)) == sinh(2)*cos(3)*exp(2)
assert trigsimp(coth(2)*cos(3)*exp(2)*sinh(2)) == cosh(2)*cos(3)*exp(2)
assert trigsimp(tanh(2)*cos(3)*exp(2)/sinh(2)) == cos(3)*exp(2)/cosh(2)
assert trigsimp(coth(2)*cos(3)*exp(2)/cosh(2)) == cos(3)*exp(2)/sinh(2)
assert trigsimp(coth(2)*cos(3)*exp(2)*tanh(2)) == cos(3)*exp(2)
def test_gruntz_hyperbolic():
assert limit(cosh(x), x, oo) == oo
assert limit(cosh(-x), x, oo) == oo
assert limit(sinh(x), x, oo) == oo
assert limit(sinh(-x), x, oo) == -oo
assert limit(2*cosh(x)*exp(x), x, oo) == oo
assert limit(2*cosh(-x)*exp(-x), x, oo) == 1
assert limit(2*sinh(x)*exp(x), x, oo) == oo
assert limit(2*sinh(-x)*exp(-x), x, oo) == -1
assert limit(tanh(x), x, oo) == 1
assert limit(tanh(-x), x, oo) == -1
assert limit(coth(x), x, oo) == 1
assert limit(coth(-x), x, oo) == -1
def test_hyperbolic_simp():
assert trigsimp(sinh(x)**2 + 1) == cosh(x)**2
assert trigsimp(cosh(x)**2 - 1) == sinh(x)**2
assert trigsimp(cosh(x)**2 - sinh(x)**2) == 1
assert trigsimp(1 - tanh(x)**2) == 1 / cosh(x)**2
assert trigsimp(1 - 1 / cosh(x)**2) == tanh(x)**2
assert trigsimp(tanh(x)**2 + 1 / cosh(x)**2) == 1
assert trigsimp(coth(x)**2 - 1) == 1 / sinh(x)**2
assert trigsimp(1 / sinh(x)**2 + 1) == 1 / tanh(x)**2
assert trigsimp(coth(x)**2 - 1 / sinh(x)**2) == 1
assert trigsimp(5 * cosh(x)**2 - 5 * sinh(x)**2) == 5
assert trigsimp(5 * cosh(x / 2)**2 -
2 * sinh(x / 2)**2) == 3 * cosh(x) / 2 + Rational(7, 2)
assert trigsimp(sinh(x) / cosh(x)) == tanh(x)
assert trigsimp(tanh(x)) == trigsimp(sinh(x) / cosh(x))
assert trigsimp(cosh(x) / sinh(x)) == 1 / tanh(x)
assert trigsimp(2 * tanh(x) * cosh(x)) == 2 * sinh(x)
assert trigsimp(coth(x)**3 * sinh(x)**3) == cosh(x)**3
assert trigsimp(y * tanh(x)**2 / sinh(x)**2) == y / cosh(x)**2
assert trigsimp(coth(x) / cosh(x)) == 1 / sinh(x)
e = 2 * cosh(x)**2 - 2 * sinh(x)**2
assert trigsimp(log(e)) == log(2)
assert trigsimp(cosh(x)**2 * cosh(y)**2 - cosh(x)**2 * sinh(y)**2 -
sinh(x)**2,
recursive=True) == 1
assert trigsimp(sinh(x)**2 * sinh(y)**2 - sinh(x)**2 * cosh(y)**2 +
cosh(x)**2,
recursive=True) == 1
assert abs(trigsimp(2.0 * cosh(x)**2 - 2.0 * sinh(x)**2) - 2.0) < 1e-10
assert trigsimp(sinh(x)**2 / cosh(x)**2) == tanh(x)**2
assert trigsimp(sinh(x)**3 / cosh(x)**3) == tanh(x)**3
assert trigsimp(sinh(x)**10 / cosh(x)**10) == tanh(x)**10
assert trigsimp(cosh(x)**3 / sinh(x)**3) == 1 / tanh(x)**3
assert trigsimp(cosh(x) / sinh(x)) == 1 / tanh(x)
assert trigsimp(cosh(x)**2 / sinh(x)**2) == 1 / tanh(x)**2
assert trigsimp(cosh(x)**10 / sinh(x)**10) == 1 / tanh(x)**10
assert trigsimp(x * cosh(x) * tanh(x)) == x * sinh(x)
assert trigsimp(-sinh(x) + cosh(x) * tanh(x)) == 0
assert tan(x) != 1 / cot(x) # cot doesn't auto-simplify
assert trigsimp(tan(x) - 1 / cot(x)) == 0
assert trigsimp(3 * tanh(x)**7 - 2 / coth(x)**7) == tanh(x)**7
def test_hyperbolic_simp():
assert trigsimp(sinh(x)**2 + 1) == cosh(x)**2
assert trigsimp(cosh(x)**2 - 1) == sinh(x)**2
assert trigsimp(cosh(x)**2 - sinh(x)**2) == 1
assert trigsimp(1 - tanh(x)**2) == 1/cosh(x)**2
assert trigsimp(1 - 1/cosh(x)**2) == tanh(x)**2
assert trigsimp(tanh(x)**2 + 1/cosh(x)**2) == 1
assert trigsimp(coth(x)**2 - 1) == 1/sinh(x)**2
assert trigsimp(1/sinh(x)**2 + 1) == 1/tanh(x)**2
assert trigsimp(coth(x)**2 - 1/sinh(x)**2) == 1
assert trigsimp(5*cosh(x)**2 - 5*sinh(x)**2) == 5
assert trigsimp(5*cosh(x/2)**2 - 2*sinh(x/2)**2) == 3*cosh(x)/2 + Rational(7, 2)
assert trigsimp(sinh(x)/cosh(x)) == tanh(x)
assert trigsimp(tanh(x)) == trigsimp(sinh(x)/cosh(x))
assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x)
assert trigsimp(2*tanh(x)*cosh(x)) == 2*sinh(x)
assert trigsimp(coth(x)**3*sinh(x)**3) == cosh(x)**3
assert trigsimp(y*tanh(x)**2/sinh(x)**2) == y/cosh(x)**2
assert trigsimp(coth(x)/cosh(x)) == 1/sinh(x)
e = 2*cosh(x)**2 - 2*sinh(x)**2
assert trigsimp(log(e)) == log(2)
assert trigsimp(cosh(x)**2*cosh(y)**2 - cosh(x)**2*sinh(y)**2 - sinh(x)**2,
recursive=True) == 1
assert trigsimp(sinh(x)**2*sinh(y)**2 - sinh(x)**2*cosh(y)**2 + cosh(x)**2,
recursive=True) == 1
assert abs(trigsimp(2.0*cosh(x)**2 - 2.0*sinh(x)**2) - 2.0) < 1e-10
assert trigsimp(sinh(x)**2/cosh(x)**2) == tanh(x)**2
assert trigsimp(sinh(x)**3/cosh(x)**3) == tanh(x)**3
assert trigsimp(sinh(x)**10/cosh(x)**10) == tanh(x)**10
assert trigsimp(cosh(x)**3/sinh(x)**3) == 1/tanh(x)**3
assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x)
assert trigsimp(cosh(x)**2/sinh(x)**2) == 1/tanh(x)**2
assert trigsimp(cosh(x)**10/sinh(x)**10) == 1/tanh(x)**10
assert trigsimp(x*cosh(x)*tanh(x)) == x*sinh(x)
assert trigsimp(-sinh(x) + cosh(x)*tanh(x)) == 0
assert tan(x) != 1/cot(x) # cot doesn't auto-simplify
assert trigsimp(tan(x) - 1/cot(x)) == 0
assert trigsimp(3*tanh(x)**7 - 2/coth(x)**7) == tanh(x)**7
def test_cosh_rewrite():
assert cosh(x).rewrite(exp) == (exp(x) + exp(-x))/2 \
== cosh(x).rewrite('tractable')
assert cosh(x).rewrite(sinh) == -I * sinh(x + I * pi / 2)
tanh_half = tanh(S.Half * x)**2
assert cosh(x).rewrite(tanh) == (1 + tanh_half) / (1 - tanh_half)
coth_half = coth(S.Half * x)**2
assert cosh(x).rewrite(coth) == (coth_half + 1) / (coth_half - 1)
def test_sinh_rewrite():
assert sinh(x).rewrite(exp) == (exp(x) - exp(-x))/2 \
== sinh(x).rewrite('tractable')
assert sinh(x).rewrite(cosh) == -I * cosh(x + I * pi / 2)
tanh_half = tanh(S.Half * x)
assert sinh(x).rewrite(tanh) == 2 * tanh_half / (1 - tanh_half**2)
coth_half = coth(S.Half * x)
assert sinh(x).rewrite(coth) == 2 * coth_half / (coth_half**2 - 1)
def test_csch_rewrite():
assert csch(x).rewrite(exp) == 1 / (exp(x)/2 - exp(-x)/2) \
== csch(x).rewrite('tractable')
assert csch(x).rewrite(cosh) == I / cosh(x + I * pi / 2)
tanh_half = tanh(S.Half * x)
assert csch(x).rewrite(tanh) == (1 - tanh_half**2) / (2 * tanh_half)
coth_half = coth(S.Half * x)
assert csch(x).rewrite(coth) == (coth_half**2 - 1) / (2 * coth_half)
def test_sech_rewrite():
assert sech(x).rewrite(exp) == 1 / (exp(x)/2 + exp(-x)/2) \
== sech(x).rewrite('tractable')
assert sech(x).rewrite(sinh) == I / sinh(x + I * pi / 2)
tanh_half = tanh(S.Half * x)**2
assert sech(x).rewrite(tanh) == (1 - tanh_half) / (1 + tanh_half)
coth_half = coth(S.Half * x)**2
assert sech(x).rewrite(coth) == (coth_half - 1) / (coth_half + 1)
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_conjugate():
a = Symbol('a', extended_real=True)
b = Symbol('b', extended_real=True)
c = Symbol('c', imaginary=True)
d = Symbol('d', imaginary=True)
z = a + I * b + c + I * d
zc = a - I * b - c + I * d
assert conjugate(z) == zc
assert conjugate(exp(z)) == exp(zc)
assert conjugate(exp(I * x)) == exp(-I * conjugate(x))
assert conjugate(z**5) == zc**5
assert conjugate(abs(x)) == abs(x)
assert conjugate(sign(z)) == sign(zc)
assert conjugate(sin(z)) == sin(zc)
assert conjugate(cos(z)) == cos(zc)
assert conjugate(tan(z)) == tan(zc)
assert conjugate(cot(z)) == cot(zc)
assert conjugate(sinh(z)) == sinh(zc)
assert conjugate(cosh(z)) == cosh(zc)
assert conjugate(tanh(z)) == tanh(zc)
assert conjugate(coth(z)) == coth(zc)
def test_trigsimp_old(capsys):
e = 2 * sin(x)**2 + 2 * cos(x)**2
assert trigsimp(e, old=True) == 2
e = 3 * tanh(x)**7 - 2 / coth(x)**7
assert trigsimp(e, method='old') == e
e = (-sin(x) + 1) / cos(x) + cos(x) / (-sin(x) + 1)
assert (trigsimp(e, method='old') == (-sin(x) + 1) / cos(x) - cos(x) /
(sin(x) - 1))
e = (-sin(x) + 1) / cos(x) + cos(x) / (-sin(x) + 1)
assert trigsimp(e, method='groebner', old=True) == 2 / cos(x)
assert trigsimp(1 / cot(x)**2, compare=True, old=True) == cot(x)**(-2)
assert capsys.readouterr().out == '\tfutrig: tan(x)**2\n'
def test_cot_rewrite():
neg_exp, pos_exp = exp(-x*I), exp(x*I)
assert cot(x).rewrite(exp) == I*(pos_exp + neg_exp)/(pos_exp - neg_exp)
assert cot(x).rewrite(sin) == 2*sin(2*x)/sin(x)**2
assert cot(x).rewrite(cos) == -cos(x)/cos(x + pi/2)
assert cot(x).rewrite(tan) == 1/tan(x)
assert (cot(sinh(x)).rewrite(exp).subs({x: 3}).evalf() ==
cot(x).rewrite(exp).subs({x: sinh(3)}).evalf())
assert (cot(cosh(x)).rewrite(exp).subs({x: 3}).evalf() ==
cot(x).rewrite(exp).subs({x: cosh(3)}).evalf())
assert (cot(tanh(x)).rewrite(exp).subs({x: 3}).evalf() ==
cot(x).rewrite(exp).subs({x: tanh(3)}).evalf())
assert (cot(coth(x)).rewrite(exp).subs({x: 3}).evalf() ==
cot(x).rewrite(exp).subs({x: coth(3)}).evalf())
assert (cot(sin(x)).rewrite(exp).subs({x: 3}).evalf() ==
cot(x).rewrite(exp).subs({x: sin(3)}).evalf())
assert (cot(tan(x)).rewrite(exp).subs({x: 3}).evalf() ==
cot(x).rewrite(exp).subs({x: tan(3)}).evalf())
assert cot(log(x)).rewrite(Pow) == -I*(x**-I + x**I)/(x**-I - x**I)
assert cot(4*pi/34).rewrite(sqrt).ratsimp() == (cos(4*pi/34)/sin(4*pi/34)).rewrite(sqrt).ratsimp()
assert cot(4*pi/17).rewrite(sqrt) == (cos(4*pi/17)/sin(4*pi/17)).rewrite(sqrt)
assert cot(pi/19).rewrite(sqrt) == cot(pi/19)
def test_conjugate():
a = Symbol("a", extended_real=True)
b = Symbol("b", extended_real=True)
c = Symbol("c", imaginary=True)
d = Symbol("d", imaginary=True)
x = Symbol('x')
z = a + I*b + c + I*d
zc = a - I*b - c + I*d
assert conjugate(z) == zc
assert conjugate(exp(z)) == exp(zc)
assert conjugate(exp(I*x)) == exp(-I*conjugate(x))
assert conjugate(z**5) == zc**5
assert conjugate(abs(x)) == abs(x)
assert conjugate(sign(z)) == sign(zc)
assert conjugate(sin(z)) == sin(zc)
assert conjugate(cos(z)) == cos(zc)
assert conjugate(tan(z)) == tan(zc)
assert conjugate(cot(z)) == cot(zc)
assert conjugate(sinh(z)) == sinh(zc)
assert conjugate(cosh(z)) == cosh(zc)
assert conjugate(tanh(z)) == tanh(zc)
assert conjugate(coth(z)) == coth(zc)
def test_cos_rewrite():
assert cos(x).rewrite(exp) == exp(I*x)/2 + exp(-I*x)/2
assert cos(x).rewrite(tan) == (1 - tan(x/2)**2)/(1 + tan(x/2)**2)
assert cos(x).rewrite(cot) == -(1 - cot(x/2)**2)/(1 + cot(x/2)**2)
assert (cos(sinh(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: sinh(3)}).evalf())
assert (cos(cosh(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: cosh(3)}).evalf())
assert (cos(tanh(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: tanh(3)}).evalf())
assert (cos(coth(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: coth(3)}).evalf())
assert (cos(sin(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: sin(3)}).evalf())
assert (cos(cos(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: cos(3)}).evalf())
assert (cos(tan(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: tan(3)}).evalf())
assert (cos(cot(x)).rewrite(exp).subs({x: 3}).evalf() ==
cos(x).rewrite(exp).subs({x: cot(3)}).evalf())
assert cos(log(x)).rewrite(Pow) == x**I/2 + x**-I/2
assert cos(x).rewrite(Pow) == cos(x)
assert cos(x).rewrite(sec) == 1/sec(x)
def test_sin_rewrite():
assert sin(x).rewrite(exp) == -I*(exp(I*x) - exp(-I*x))/2
assert sin(x).rewrite(tan) == 2*tan(x/2)/(1 + tan(x/2)**2)
assert sin(x).rewrite(cot) == 2*cot(x/2)/(1 + cot(x/2)**2)
assert (sin(sinh(x)).rewrite(exp).subs({x: 3}).evalf() ==
sin(x).rewrite(exp).subs({x: sinh(3)}).evalf())
assert (sin(cosh(x)).rewrite(exp).subs({x: 3}).evalf() ==
sin(x).rewrite(exp).subs({x: cosh(3)}).evalf())
assert (sin(tanh(x)).rewrite(exp).subs({x: 3}).evalf() ==
sin(x).rewrite(exp).subs({x: tanh(3)}).evalf())
assert (sin(coth(x)).rewrite(exp).subs({x: 3}).evalf() ==
sin(x).rewrite(exp).subs({x: coth(3)}).evalf())
assert (sin(sin(x)).rewrite(exp).subs({x: 3}).evalf() ==
sin(x).rewrite(exp).subs({x: sin(3)}).evalf())
assert (sin(cos(x)).rewrite(exp).subs({x: 3}).evalf() ==
sin(x).rewrite(exp).subs({x: cos(3)}).evalf())
assert (sin(tan(x)).rewrite(exp).subs({x: 3}).evalf() ==
sin(x).rewrite(exp).subs({x: tan(3)}).evalf())
assert (sin(cot(x)).rewrite(exp).subs({x: 3}).evalf() ==
sin(x).rewrite(exp).subs({x: cot(3)}).evalf())
assert sin(log(x)).rewrite(Pow) == I*x**-I / 2 - I*x**I / 2
assert sin(x).rewrite(Pow) == sin(x) # issue sympy/sympy#7171
assert sin(x).rewrite(csc) == 1/csc(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_hyper_as_trig():
eq = sinh(x)**2 + cosh(x)**2
t, f = hyper_as_trig(eq)
assert f(fu(t)) == cosh(2*x)
e, f = hyper_as_trig(tanh(x + y))
assert f(TR12(e)) == (tanh(x) + tanh(y))/(tanh(x)*tanh(y) + 1)
d = Dummy()
assert o(sinh(x), d) == I*sin(x*d)
assert o(tanh(x), d) == I*tan(x*d)
assert o(coth(x), d) == cot(x*d)/I
assert o(cosh(x), d) == cos(x*d)
for func in (sinh, cosh, tanh, coth):
h = func(pi)
assert i(o(h, d), d) == h
# /!\ the _osborne functions are not meant to work
# in the o(i(trig, d), d) direction so we just check
# that they work as they are supposed to work
assert i(cos(x*y), y) == cosh(x)
assert i(sin(x*y), y) == sinh(x)/I
assert i(tan(x*y), y) == tanh(x)/I
assert i(cot(x*y), y) == coth(x)*I
assert i(sec(x*y), y) == 1/cosh(x)
assert i(csc(x*y), y) == I/sinh(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_complex():
a, b = symbols('a,b', extended_real=True)
z = a + b*I
for func in [sinh, cosh, tanh, coth, sech, csch]:
assert func(z).conjugate() == func(a - b*I)
for deep in [True, False]:
assert sinh(z).expand(
complex=True, deep=deep) == sinh(a)*cos(b) + I*cosh(a)*sin(b)
assert cosh(z).expand(
complex=True, deep=deep) == cosh(a)*cos(b) + I*sinh(a)*sin(b)
assert tanh(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
a)/(cos(b)**2 + sinh(a)**2) + I*sin(b)*cos(b)/(cos(b)**2 + sinh(a)**2)
assert coth(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
a)/(sin(b)**2 + sinh(a)**2) - I*sin(b)*cos(b)/(sin(b)**2 + sinh(a)**2)
assert csch(z).expand(complex=True, deep=deep) == cos(b) * sinh(a) / (sin(b)**2
* cosh(a)**2 + cos(b)**2 * sinh(a)**2) - I*sin(b) * cosh(a) / (sin(b)**2
* cosh(a)**2 + cos(b)**2 * sinh(a)**2)
assert sech(z).expand(complex=True, deep=deep) == cos(b) * cosh(a) / (sin(b)**2
* sinh(a)**2 + cos(b)**2 * cosh(a)**2) - I*sin(b) * sinh(a) / (sin(b)**2
* sinh(a)**2 + cos(b)**2 * cosh(a)**2)
def test_complex():
a, b = symbols('a,b', extended_real=True)
z = a + b*I
for func in [sinh, cosh, tanh, coth, sech, csch]:
assert func(z).conjugate() == func(a - b*I)
for deep in [True, False]:
assert sinh(z).expand(
complex=True, deep=deep) == sinh(a)*cos(b) + I*cosh(a)*sin(b)
assert cosh(z).expand(
complex=True, deep=deep) == cosh(a)*cos(b) + I*sinh(a)*sin(b)
assert tanh(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
a)/(cos(b)**2 + sinh(a)**2) + I*sin(b)*cos(b)/(cos(b)**2 + sinh(a)**2)
assert coth(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
a)/(sin(b)**2 + sinh(a)**2) - I*sin(b)*cos(b)/(sin(b)**2 + sinh(a)**2)
assert csch(z).expand(complex=True, deep=deep) == cos(b) * sinh(a) / (sin(b)**2
* cosh(a)**2 + cos(b)**2 * sinh(a)**2) - I*sin(b) * cosh(a) / (sin(b)**2
* cosh(a)**2 + cos(b)**2 * sinh(a)**2)
assert sech(z).expand(complex=True, deep=deep) == cos(b) * cosh(a) / (sin(b)**2
* sinh(a)**2 + cos(b)**2 * cosh(a)**2) - I*sin(b) * sinh(a) / (sin(b)**2
* sinh(a)**2 + cos(b)**2 * cosh(a)**2)
def test_coth():
k = Symbol('k', integer=True)
assert coth(nan) == nan
assert coth(zoo) == nan
assert coth(oo) == 1
assert coth(-oo) == -1
assert coth(0) == coth(0)
assert coth(0) == zoo
assert coth(1) == coth(1)
assert coth(-1) == -coth(1)
assert coth(x) == coth(x)
assert coth(-x) == -coth(x)
assert coth(pi*I) == -I*cot(pi)
assert coth(-pi*I) == cot(pi)*I
assert coth(2**1024 * E) == coth(2**1024 * E)
assert coth(-2**1024 * E) == -coth(2**1024 * E)
assert coth(pi*I) == -I*cot(pi)
assert coth(-pi*I) == I*cot(pi)
assert coth(2*pi*I) == -I*cot(2*pi)
assert coth(-2*pi*I) == I*cot(2*pi)
assert coth(-3*10**73*pi*I) == I*cot(3*10**73*pi)
assert coth(7*10**103*pi*I) == -I*cot(7*10**103*pi)
assert coth(pi*I/2) == 0
assert coth(-pi*I/2) == 0
assert coth(5*pi*I/2) == 0
assert coth(7*pi*I/2) == 0
assert coth(pi*I/3) == -I/sqrt(3)
assert coth(-2*pi*I/3) == -I/sqrt(3)
assert coth(pi*I/4) == -I
assert coth(-pi*I/4) == I
assert coth(17*pi*I/4) == -I
assert coth(-3*pi*I/4) == -I
assert coth(pi*I/6) == -sqrt(3)*I
assert coth(-pi*I/6) == sqrt(3)*I
assert coth(7*pi*I/6) == -sqrt(3)*I
assert coth(-5*pi*I/6) == -sqrt(3)*I
assert coth(pi*I/105) == -cot(pi/105)*I
assert coth(-pi*I/105) == cot(pi/105)*I
assert coth(2 + 3*I) == coth(2 + 3*I)
assert coth(x*I) == -cot(x)*I
assert coth(k*pi*I) == -cot(k*pi)*I
assert coth(17*k*pi*I) == -cot(17*k*pi)*I
assert coth(k*pi*I) == -cot(k*pi)*I
pytest.raises(ArgumentIndexError, lambda: coth(x).fdiff(2))
a, b = symbols('a b', extended_real=True)
z = a + b*I
for deep in [True, False]:
d = sinh(a)**2 + sin(b)**2
assert coth(z).as_real_imag(deep=deep) == (sinh(a)*cosh(a)/d,
-sin(b)*cos(b)/d)
assert coth(a).as_real_imag(deep=deep) == (coth(a), 0)
def test_coth_series():
assert coth(x).series(x, 0, 8) == \
1/x + x/3 - x**3/45 + 2*x**5/945 - x**7/4725 + O(x**8)
def test_sinh_rewrite():
assert sinh(x).rewrite(exp) == (exp(x) - exp(-x))/2 \
== sinh(x).rewrite('tractable')
assert sinh(x).rewrite(cosh) == -I*cosh(x + I*pi/2)
assert sinh(x).rewrite(tanh) == 2*tanh(x/2)/(1 - tanh(x/2)**2)
assert sinh(x).rewrite(coth) == 2*coth(x/2)/(coth(x/2)**2 - 1)
def test_coth_series():
assert coth(x).series(x, 0, 8) == \
1/x + x/3 - x**3/45 + 2*x**5/945 - x**7/4725 + O(x**8)
def test_csch_rewrite():
assert csch(x).rewrite(exp) == 1 / (exp(x)/2 - exp(-x)/2) \
== csch(x).rewrite('tractable')
assert csch(x).rewrite(cosh) == I / cosh(x + I * pi / 2)
assert csch(x).rewrite(tanh) == (1 - tanh(x / 2)**2) / (2 * tanh(x / 2))
assert csch(x).rewrite(coth) == (coth(x / 2)**2 - 1) / (2 * coth(x / 2))
def test_tanh_rewrite():
assert tanh(x).rewrite(exp) == (exp(x) - exp(-x))/(exp(x) + exp(-x)) \
== tanh(x).rewrite('tractable')
assert tanh(x).rewrite(sinh) == I * sinh(x) / sinh(I * pi / 2 - x)
assert tanh(x).rewrite(cosh) == I * cosh(I * pi / 2 - x) / cosh(x)
assert tanh(x).rewrite(coth) == 1 / coth(x)
def test_cosh_rewrite():
assert cosh(x).rewrite(exp) == (exp(x) + exp(-x))/2 \
== cosh(x).rewrite('tractable')
assert cosh(x).rewrite(sinh) == -I * sinh(x + I * pi / 2)
assert cosh(x).rewrite(tanh) == (1 + tanh(x / 2)**2) / (1 - tanh(x / 2)**2)
assert cosh(x).rewrite(coth) == (coth(x / 2)**2 + 1) / (coth(x / 2)**2 - 1)
def test_sinh_rewrite():
assert sinh(x).rewrite(exp) == (exp(x) - exp(-x))/2 \
== sinh(x).rewrite('tractable')
assert sinh(x).rewrite(cosh) == -I * cosh(x + I * pi / 2)
assert sinh(x).rewrite(tanh) == 2 * tanh(x / 2) / (1 - tanh(x / 2)**2)
assert sinh(x).rewrite(coth) == 2 * coth(x / 2) / (coth(x / 2)**2 - 1)
def test_coth_rewrite():
assert coth(x).rewrite(exp) == (exp(x) + exp(-x))/(exp(x) - exp(-x)) \
== coth(x).rewrite('tractable')
assert coth(x).rewrite(sinh) == -I*sinh(I*pi/2 - x)/sinh(x)
assert coth(x).rewrite(cosh) == -I*cosh(x)/cosh(I*pi/2 - x)
assert coth(x).rewrite(tanh) == 1/tanh(x)
def test_csch_rewrite():
assert csch(x).rewrite(exp) == 1 / (exp(x)/2 - exp(-x)/2) \
== csch(x).rewrite('tractable')
assert csch(x).rewrite(cosh) == I/cosh(x + I*pi/2)
assert csch(x).rewrite(tanh) == (1 - tanh(x/2)**2)/(2*tanh(x/2))
assert csch(x).rewrite(coth) == (coth(x/2)**2 - 1)/(2*coth(x/2))
def test_sech_rewrite():
assert sech(x).rewrite(exp) == 1 / (exp(x)/2 + exp(-x)/2) \
== sech(x).rewrite('tractable')
assert sech(x).rewrite(sinh) == I / sinh(x + I * pi / 2)
assert sech(x).rewrite(tanh) == (1 - tanh(x / 2)**2) / (1 + tanh(x / 2)**2)
assert sech(x).rewrite(coth) == (coth(x / 2)**2 - 1) / (coth(x / 2)**2 + 1)
def test_cosh_rewrite():
assert cosh(x).rewrite(exp) == (exp(x) + exp(-x))/2 \
== cosh(x).rewrite('tractable')
assert cosh(x).rewrite(sinh) == -I*sinh(x + I*pi/2)
assert cosh(x).rewrite(tanh) == (1 + tanh(x/2)**2)/(1 - tanh(x/2)**2)
assert cosh(x).rewrite(coth) == (coth(x/2)**2 + 1)/(coth(x/2)**2 - 1)
def test_sech_rewrite():
assert sech(x).rewrite(exp) == 1 / (exp(x)/2 + exp(-x)/2) \
== sech(x).rewrite('tractable')
assert sech(x).rewrite(sinh) == I/sinh(x + I*pi/2)
assert sech(x).rewrite(tanh) == (1 - tanh(x/2)**2)/(1 + tanh(x/2)**2)
assert sech(x).rewrite(coth) == (coth(x/2)**2 - 1)/(coth(x/2)**2 + 1)
def test_coth():
k = Symbol('k', integer=True)
assert coth(nan) == nan
assert coth(zoo) == nan
assert coth(oo) == 1
assert coth(-oo) == -1
assert coth(0) == coth(0)
assert coth(0) == zoo
assert coth(1) == coth(1)
assert coth(-1) == -coth(1)
assert coth(x) == coth(x)
assert coth(-x) == -coth(x)
assert coth(pi * I) == -I * cot(pi)
assert coth(-pi * I) == cot(pi) * I
assert coth(2**1024 * E) == coth(2**1024 * E)
assert coth(-2**1024 * E) == -coth(2**1024 * E)
assert coth(pi * I) == -I * cot(pi)
assert coth(-pi * I) == I * cot(pi)
assert coth(2 * pi * I) == -I * cot(2 * pi)
assert coth(-2 * pi * I) == I * cot(2 * pi)
assert coth(-3 * 10**73 * pi * I) == I * cot(3 * 10**73 * pi)
assert coth(7 * 10**103 * pi * I) == -I * cot(7 * 10**103 * pi)
assert coth(pi * I / 2) == 0
assert coth(-pi * I / 2) == 0
assert coth(5 * pi * I / 2) == 0
assert coth(7 * pi * I / 2) == 0
assert coth(pi * I / 3) == -I / sqrt(3)
assert coth(-2 * pi * I / 3) == -I / sqrt(3)
assert coth(pi * I / 4) == -I
assert coth(-pi * I / 4) == I
assert coth(17 * pi * I / 4) == -I
assert coth(-3 * pi * I / 4) == -I
assert coth(pi * I / 6) == -sqrt(3) * I
assert coth(-pi * I / 6) == sqrt(3) * I
assert coth(7 * pi * I / 6) == -sqrt(3) * I
assert coth(-5 * pi * I / 6) == -sqrt(3) * I
assert coth(pi * I / 105) == -cot(pi / 105) * I
assert coth(-pi * I / 105) == cot(pi / 105) * I
assert coth(2 + 3 * I) == coth(2 + 3 * I)
assert coth(x * I) == -cot(x) * I
assert coth(k * pi * I) == -cot(k * pi) * I
assert coth(17 * k * pi * I) == -cot(17 * k * pi) * I
assert coth(k * pi * I) == -cot(k * pi) * I
pytest.raises(ArgumentIndexError, lambda: coth(x).fdiff(2))
a, b = symbols('a b', extended_real=True)
z = a + b * I
for deep in [True, False]:
d = sinh(a)**2 + sin(b)**2
assert coth(z).as_real_imag(deep=deep) == (sinh(a) * cosh(a) / d,
-sin(b) * cos(b) / d)
assert coth(a).as_real_imag(deep=deep) == (coth(a), 0)
