def test_hyper_as_trig():
from sympy.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)
assert o(sech(x), d) == sec(x*d)
assert o(csch(x), d) == csc(x*d)/I
for func in (sinh, cosh, tanh, coth, sech, csch):
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 + z), y) == cosh(x + z*I)
assert i(sin(x*y + z), y) == sinh(x + z*I)/I
assert i(tan(x*y + z), y) == tanh(x + z*I)/I
assert i(cot(x*y + z), y) == coth(x + z*I)*I
assert i(sec(x*y + z), y) == sech(x + z*I)
assert i(csc(x*y + z), y) == csch(x + z*I)*I
def test_simplifications():
x = Symbol('x')
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():
x = Symbol('x')
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 + S.Pi / 2) / cos(x)
assert tan(x).rewrite(cot) == 1 / cot(x)
assert tan(sinh(x)).rewrite(exp).subs(x,
3).n() == tan(x).rewrite(exp).subs(
x, sinh(3)).n()
assert tan(cosh(x)).rewrite(exp).subs(x,
3).n() == tan(x).rewrite(exp).subs(
x, cosh(3)).n()
assert tan(tanh(x)).rewrite(exp).subs(x,
3).n() == tan(x).rewrite(exp).subs(
x, tanh(3)).n()
assert tan(coth(x)).rewrite(exp).subs(x,
3).n() == tan(x).rewrite(exp).subs(
x, coth(3)).n()
assert tan(sin(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(
x, sin(3)).n()
assert tan(cos(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(
x, cos(3)).n()
assert tan(tan(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(
x, tan(3)).n()
assert tan(cot(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(
x, cot(3)).n()
assert tan(log(x)).rewrite(Pow) == I * (x**-I - x**I) / (x**-I + x**I)
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_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 + S.Pi / 2) / cos(x)
assert tan(x).rewrite(cot) == 1 / cot(x)
assert tan(sinh(x)).rewrite(exp).subs(x,
3).n() == tan(x).rewrite(exp).subs(
x, sinh(3)).n()
assert tan(cosh(x)).rewrite(exp).subs(x,
3).n() == tan(x).rewrite(exp).subs(
x, cosh(3)).n()
assert tan(tanh(x)).rewrite(exp).subs(x,
3).n() == tan(x).rewrite(exp).subs(
x, tanh(3)).n()
assert tan(coth(x)).rewrite(exp).subs(x,
3).n() == tan(x).rewrite(exp).subs(
x, coth(3)).n()
assert tan(sin(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(
x, sin(3)).n()
assert tan(cos(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(
x, cos(3)).n()
assert tan(tan(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(
x, tan(3)).n()
assert tan(cot(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(
x, cot(3)).n()
assert tan(log(x)).rewrite(Pow) == I * (x**-I - x**I) / (x**-I + x**I)
assert 0 == (cos(pi / 15) * tan(pi / 15) - sin(pi / 15)).rewrite(pow)
assert tan(pi / 19).rewrite(pow) == tan(pi / 19)
assert tan(8 * pi / 19).rewrite(sqrt) == tan(8 * pi / 19)
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).n() == sin(x).rewrite(exp).subs(
x, sinh(3)).n()
assert sin(cosh(x)).rewrite(exp).subs(x,
3).n() == sin(x).rewrite(exp).subs(
x, cosh(3)).n()
assert sin(tanh(x)).rewrite(exp).subs(x,
3).n() == sin(x).rewrite(exp).subs(
x, tanh(3)).n()
assert sin(coth(x)).rewrite(exp).subs(x,
3).n() == sin(x).rewrite(exp).subs(
x, coth(3)).n()
assert sin(sin(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(
x, sin(3)).n()
assert sin(cos(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(
x, cos(3)).n()
assert sin(tan(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(
x, tan(3)).n()
assert sin(cot(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(
x, cot(3)).n()
assert sin(log(x)).rewrite(Pow) == I * x**-I / 2 - I * x**I / 2
assert sin(x).rewrite(csc) == 1 / csc(x)
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 + S.Pi / 2)
assert cot(x).rewrite(tan) == 1 / tan(x)
assert cot(sinh(x)).rewrite(exp).subs(x,
3).n() == cot(x).rewrite(exp).subs(
x, sinh(3)).n()
assert cot(cosh(x)).rewrite(exp).subs(x,
3).n() == cot(x).rewrite(exp).subs(
x, cosh(3)).n()
assert cot(tanh(x)).rewrite(exp).subs(x,
3).n() == cot(x).rewrite(exp).subs(
x, tanh(3)).n()
assert cot(coth(x)).rewrite(exp).subs(x,
3).n() == cot(x).rewrite(exp).subs(
x, coth(3)).n()
assert cot(sin(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(
x, sin(3)).n()
assert cot(tan(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(
x, tan(3)).n()
assert cot(log(x)).rewrite(Pow) == -I * (x**-I + x**I) / (x**-I - x**I)
assert cot(4 * pi / 15).rewrite(pow) == (cos(4 * pi / 15) /
sin(4 * pi / 15)).rewrite(pow)
assert cot(pi / 19).rewrite(pow) == cot(pi / 19)
assert cot(pi / 19).rewrite(sqrt) == cot(pi / 19)
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).n() == cos(x).rewrite(exp).subs(
x, sinh(3)).n()
assert cos(cosh(x)).rewrite(exp).subs(x,
3).n() == cos(x).rewrite(exp).subs(
x, cosh(3)).n()
assert cos(tanh(x)).rewrite(exp).subs(x,
3).n() == cos(x).rewrite(exp).subs(
x, tanh(3)).n()
assert cos(coth(x)).rewrite(exp).subs(x,
3).n() == cos(x).rewrite(exp).subs(
x, coth(3)).n()
assert cos(sin(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(
x, sin(3)).n()
assert cos(cos(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(
x, cos(3)).n()
assert cos(tan(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(
x, tan(3)).n()
assert cos(cot(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(
x, cot(3)).n()
assert cos(log(x)).rewrite(Pow) == x**I / 2 + x**-I / 2
assert cos(x).rewrite(sec) == 1 / sec(x)
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_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 + S.Pi/2)/cos(x)
assert tan(x).rewrite(cot) == 1/cot(x)
assert tan(sinh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n()
assert tan(cosh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n()
assert tan(tanh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n()
assert tan(coth(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n()
assert tan(sin(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n()
assert tan(cos(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n()
assert tan(tan(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n()
assert tan(cot(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n()
assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
assert 0 == (cos(pi/15)*tan(pi/15) - sin(pi/15)).rewrite(pow)
assert tan(pi/19).rewrite(pow) == tan(pi/19)
assert tan(8*pi/19).rewrite(sqrt) == tan(8*pi/19)
def test_hyper_as_trig():
from sympy.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_hyperbolic_simp():
x, y = symbols('x,y')
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)
for a in (pi / 6 * I, pi / 4 * I, pi / 3 * I):
assert trigsimp(sinh(a) * cosh(x) + cosh(a) * sinh(x)) == sinh(x + a)
assert trigsimp(-sinh(a) * cosh(x) + cosh(a) * sinh(x)) == sinh(x - a)
e = 2 * cosh(x)**2 - 2 * sinh(x)**2
assert trigsimp(log(e)) == log(2)
# issue 19535:
assert trigsimp(sqrt(cosh(x)**2 - 1)) == sqrt(sinh(x)**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_simplifications():
x = Symbol("x")
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_cot():
assert cot(nan) == nan
assert cot.nargs == FiniteSet(1)
assert cot(oo*I) == -I
assert cot(-oo*I) == I
assert cot(0) == zoo
assert cot(2*pi) == zoo
assert cot(acot(x)) == x
assert cot(atan(x)) == 1 / x
assert cot(asin(x)) == sqrt(1 - x**2) / x
assert cot(acos(x)) == x / sqrt(1 - x**2)
assert cot(atan2(y, x)) == x/y
assert cot(pi*I) == -coth(pi)*I
assert cot(-pi*I) == coth(pi)*I
assert cot(-2*I) == coth(2)*I
assert cot(pi) == cot(2*pi) == cot(3*pi)
assert cot(-pi) == cot(-2*pi) == cot(-3*pi)
assert cot(pi/2) == 0
assert cot(-pi/2) == 0
assert cot(5*pi/2) == 0
assert cot(7*pi/2) == 0
assert cot(pi/3) == 1/sqrt(3)
assert cot(-2*pi/3) == 1/sqrt(3)
assert cot(pi/4) == S.One
assert cot(-pi/4) == -S.One
assert cot(17*pi/4) == S.One
assert cot(-3*pi/4) == S.One
assert cot(pi/6) == sqrt(3)
assert cot(-pi/6) == -sqrt(3)
assert cot(7*pi/6) == sqrt(3)
assert cot(-5*pi/6) == sqrt(3)
assert cot(x*I) == -coth(x)*I
assert cot(k*pi*I) == -coth(k*pi)*I
assert cot(r).is_real is True
assert cot(a).is_algebraic is None
assert cot(na).is_algebraic is False
assert cot(10*pi/7) == cot(3*pi/7)
assert cot(11*pi/7) == -cot(3*pi/7)
assert cot(-11*pi/7) == cot(3*pi/7)
assert cot(x).is_finite is None
assert cot(r).is_finite is None
i = Symbol('i', imaginary=True)
assert cot(i).is_finite is True
assert cot(x).subs(x, 3*pi) == zoo
def test_cot():
assert cot(nan) == nan
assert cot.nargs == FiniteSet(1)
assert cot(oo*I) == -I
assert cot(-oo*I) == I
assert cot(0) == zoo
assert cot(2*pi) == zoo
assert cot(acot(x)) == x
assert cot(atan(x)) == 1 / x
assert cot(asin(x)) == sqrt(1 - x**2) / x
assert cot(acos(x)) == x / sqrt(1 - x**2)
assert cot(atan2(y, x)) == x/y
assert cot(pi*I) == -coth(pi)*I
assert cot(-pi*I) == coth(pi)*I
assert cot(-2*I) == coth(2)*I
assert cot(pi) == cot(2*pi) == cot(3*pi)
assert cot(-pi) == cot(-2*pi) == cot(-3*pi)
assert cot(pi/2) == 0
assert cot(-pi/2) == 0
assert cot(5*pi/2) == 0
assert cot(7*pi/2) == 0
assert cot(pi/3) == 1/sqrt(3)
assert cot(-2*pi/3) == 1/sqrt(3)
assert cot(pi/4) == S.One
assert cot(-pi/4) == -S.One
assert cot(17*pi/4) == S.One
assert cot(-3*pi/4) == S.One
assert cot(pi/6) == sqrt(3)
assert cot(-pi/6) == -sqrt(3)
assert cot(7*pi/6) == sqrt(3)
assert cot(-5*pi/6) == sqrt(3)
assert cot(x*I) == -coth(x)*I
assert cot(k*pi*I) == -coth(k*pi)*I
assert cot(r).is_real is True
assert cot(a).is_algebraic is None
assert cot(na).is_algebraic is False
assert cot(10*pi/7) == cot(3*pi/7)
assert cot(11*pi/7) == -cot(3*pi/7)
assert cot(-11*pi/7) == cot(3*pi/7)
assert cot(x).is_finite is None
assert cot(r).is_finite is None
i = Symbol('i', imaginary=True)
assert cot(i).is_finite is True
assert cot(x).subs(x, 3*pi) == zoo
def test_cot():
x, y = symbols('x,y')
r = Symbol('r', real=True)
k = Symbol('k', integer=True)
assert cot(nan) == nan
assert cot(oo*I) == -I
assert cot(-oo*I) == I
assert cot(0) == zoo
assert cot(2*pi) == zoo
assert cot(acot(x)) == x
assert cot(atan(x)) == 1 / x
assert cot(asin(x)) == sqrt(1 - x**2) / x
assert cot(acos(x)) == x / sqrt(1 - x**2)
assert cot(atan2(y, x)) == x/y
assert cot(pi*I) == -coth(pi)*I
assert cot(-pi*I) == coth(pi)*I
assert cot(-2*I) == coth(2)*I
assert cot(pi) == cot(2*pi) == cot(3*pi)
assert cot(-pi) == cot(-2*pi) == cot(-3*pi)
assert cot(pi/2) == 0
assert cot(-pi/2) == 0
assert cot(5*pi/2) == 0
assert cot(7*pi/2) == 0
assert cot(pi/3) == 1/sqrt(3)
assert cot(-2*pi/3) == 1/sqrt(3)
assert cot(pi/4) == S.One
assert cot(-pi/4) == -S.One
assert cot(17*pi/4) == S.One
assert cot(-3*pi/4) == S.One
assert cot(pi/6) == sqrt(3)
assert cot(-pi/6) == -sqrt(3)
assert cot(7*pi/6) == sqrt(3)
assert cot(-5*pi/6) == sqrt(3)
assert cot(x*I) == -coth(x)*I
assert cot(k*pi*I) == -coth(k*pi)*I
assert cot(r).is_real is True
assert cot(10*pi/7) == cot(3*pi/7)
assert cot(11*pi/7) == -cot(3*pi/7)
assert cot(-11*pi/7) == cot(3*pi/7)
def test_cot():
x, y = symbols('x,y')
r = Symbol('r', real=True)
k = Symbol('k', integer=True)
assert cot(nan) == nan
assert cot(oo * I) == -I
assert cot(-oo * I) == I
assert cot(0) == zoo
assert cot(2 * pi) == zoo
assert cot(acot(x)) == x
assert cot(atan(x)) == 1 / x
assert cot(asin(x)) == sqrt(1 - x**2) / x
assert cot(acos(x)) == x / sqrt(1 - x**2)
assert cot(atan2(y, x)) == x / y
assert cot(pi * I) == -coth(pi) * I
assert cot(-pi * I) == coth(pi) * I
assert cot(-2 * I) == coth(2) * I
assert cot(pi) == cot(2 * pi) == cot(3 * pi)
assert cot(-pi) == cot(-2 * pi) == cot(-3 * pi)
assert cot(pi / 2) == 0
assert cot(-pi / 2) == 0
assert cot(5 * pi / 2) == 0
assert cot(7 * pi / 2) == 0
assert cot(pi / 3) == 1 / sqrt(3)
assert cot(-2 * pi / 3) == 1 / sqrt(3)
assert cot(pi / 4) == S.One
assert cot(-pi / 4) == -S.One
assert cot(17 * pi / 4) == S.One
assert cot(-3 * pi / 4) == S.One
assert cot(pi / 6) == sqrt(3)
assert cot(-pi / 6) == -sqrt(3)
assert cot(7 * pi / 6) == sqrt(3)
assert cot(-5 * pi / 6) == sqrt(3)
assert cot(x * I) == -coth(x) * I
assert cot(k * pi * I) == -coth(k * pi) * I
assert cot(r).is_real == True
assert cot(10 * pi / 7) == cot(3 * pi / 7)
assert cot(11 * pi / 7) == -cot(3 * pi / 7)
assert cot(-11 * pi / 7) == cot(3 * pi / 7)
def test_hyperbolic_simp():
x, y = symbols('x,y')
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 + S(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)
for a in (pi/6*I, pi/4*I, pi/3*I):
assert trigsimp(sinh(a)*cosh(x) + cosh(a)*sinh(x)) == sinh(x + a)
assert trigsimp(-sinh(a)*cosh(x) + cosh(a)*sinh(x)) == sinh(x - a)
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_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)
def test_hyperbolic_simp():
x, y = symbols('x,y')
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 + S(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_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).n() == sin(x).rewrite(exp).subs(x, sinh(3)).n()
assert sin(cosh(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cosh(3)).n()
assert sin(tanh(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tanh(3)).n()
assert sin(coth(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, coth(3)).n()
assert sin(sin(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sin(3)).n()
assert sin(cos(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cos(3)).n()
assert sin(tan(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tan(3)).n()
assert sin(cot(x)).rewrite(exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cot(3)).n()
assert sin(log(x)).rewrite(Pow) == I * x ** -I / 2 - I * x ** I / 2
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).n() == cos(x).rewrite(exp).subs(x, sinh(3)).n()
assert cos(cosh(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cosh(3)).n()
assert cos(tanh(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tanh(3)).n()
assert cos(coth(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, coth(3)).n()
assert cos(sin(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sin(3)).n()
assert cos(cos(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cos(3)).n()
assert cos(tan(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tan(3)).n()
assert cos(cot(x)).rewrite(exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cot(3)).n()
assert cos(log(x)).rewrite(Pow) == x ** I / 2 + x ** -I / 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_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_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_cot_rewrite():
x = Symbol('x')
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 + S.Pi/2)
assert cot(x).rewrite(tan) == 1/tan(x)
assert cot(sinh(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, sinh(3)).n()
assert cot(cosh(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, cosh(3)).n()
assert cot(tanh(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, tanh(3)).n()
assert cot(coth(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, coth(3)).n()
assert cot(sin(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, sin(3)).n()
assert cot(tan(x)).rewrite(exp).subs(x, 3).n() == cot(x).rewrite(exp).subs(x, tan(3)).n()
assert cot(log(x)).rewrite(Pow) == -I*(x**-I + x**I)/(x**-I - x**I)
def test_cot():
assert cot(nan) == nan
assert cot.nargs == FiniteSet(1)
assert cot(oo*I) == -I
assert cot(-oo*I) == I
assert cot(0) == zoo
assert cot(2*pi) == zoo
assert cot(acot(x)) == x
assert cot(atan(x)) == 1 / x
assert cot(asin(x)) == sqrt(1 - x**2) / x
assert cot(acos(x)) == x / sqrt(1 - x**2)
assert cot(atan2(y, x)) == x/y
assert cot(pi*I) == -coth(pi)*I
assert cot(-pi*I) == coth(pi)*I
assert cot(-2*I) == coth(2)*I
assert cot(pi) == cot(2*pi) == cot(3*pi)
assert cot(-pi) == cot(-2*pi) == cot(-3*pi)
assert cot(pi/2) == 0
assert cot(-pi/2) == 0
assert cot(5*pi/2) == 0
assert cot(7*pi/2) == 0
assert cot(pi/3) == 1/sqrt(3)
assert cot(-2*pi/3) == 1/sqrt(3)
assert cot(pi/4) == S.One
assert cot(-pi/4) == -S.One
assert cot(17*pi/4) == S.One
assert cot(-3*pi/4) == S.One
assert cot(pi/6) == sqrt(3)
assert cot(-pi/6) == -sqrt(3)
assert cot(7*pi/6) == sqrt(3)
assert cot(-5*pi/6) == sqrt(3)
assert cot(x*I) == -coth(x)*I
assert cot(k*pi*I) == -coth(k*pi)*I
assert cot(r).is_real is True
assert cot(10*pi/7) == cot(3*pi/7)
assert cot(11*pi/7) == -cot(3*pi/7)
assert cot(-11*pi/7) == cot(3*pi/7)
def test_cot():
assert cot(nan) == nan
assert cot.nargs == FiniteSet(1)
assert cot(oo*I) == -I
assert cot(-oo*I) == I
assert cot(0) == zoo
assert cot(2*pi) == zoo
assert cot(acot(x)) == x
assert cot(atan(x)) == 1 / x
assert cot(asin(x)) == sqrt(1 - x**2) / x
assert cot(acos(x)) == x / sqrt(1 - x**2)
assert cot(atan2(y, x)) == x/y
assert cot(pi*I) == -coth(pi)*I
assert cot(-pi*I) == coth(pi)*I
assert cot(-2*I) == coth(2)*I
assert cot(pi) == cot(2*pi) == cot(3*pi)
assert cot(-pi) == cot(-2*pi) == cot(-3*pi)
assert cot(pi/2) == 0
assert cot(-pi/2) == 0
assert cot(5*pi/2) == 0
assert cot(7*pi/2) == 0
assert cot(pi/3) == 1/sqrt(3)
assert cot(-2*pi/3) == 1/sqrt(3)
assert cot(pi/4) == S.One
assert cot(-pi/4) == -S.One
assert cot(17*pi/4) == S.One
assert cot(-3*pi/4) == S.One
assert cot(pi/6) == sqrt(3)
assert cot(-pi/6) == -sqrt(3)
assert cot(7*pi/6) == sqrt(3)
assert cot(-5*pi/6) == sqrt(3)
assert cot(x*I) == -coth(x)*I
assert cot(k*pi*I) == -coth(k*pi)*I
assert cot(r).is_real is True
assert cot(10*pi/7) == cot(3*pi/7)
assert cot(11*pi/7) == -cot(3*pi/7)
assert cot(-11*pi/7) == cot(3*pi/7)
def test_sign_assumptions():
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
assert sinh(n).is_negative is True
assert sinh(p).is_positive is True
assert cosh(n).is_positive is True
assert cosh(p).is_positive is True
assert tanh(n).is_negative is True
assert tanh(p).is_positive is True
assert csch(n).is_negative is True
assert csch(p).is_positive is True
assert sech(n).is_positive is True
assert sech(p).is_positive is True
assert coth(n).is_negative is True
assert coth(p).is_positive is True
def test_sign_assumptions():
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
assert sinh(n).is_negative is True
assert sinh(p).is_positive is True
assert cosh(n).is_positive is True
assert cosh(p).is_positive is True
assert tanh(n).is_negative is True
assert tanh(p).is_positive is True
assert csch(n).is_negative is True
assert csch(p).is_positive is True
assert sech(n).is_positive is True
assert sech(p).is_positive is True
assert coth(n).is_negative is True
assert coth(p).is_positive is True
def FJC(F):
EEDss = []
for Fext in force:
x = Lss * (coth(Fext * b / 4.1) - 4.1 / (Fext * b)) * (1 + Fext / Sss)
EEDss.append(x)
EEDss = np.array(EEDss)
return (EEDss)
def test_complex():
a, b = symbols("a,b", 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_hyper_as_trig():
from sympy.simplify.fu import _osborne, _osbornei
eq = sinh(x)**2 + cosh(x)**2
t, f = hyper_as_trig(eq)
assert f(fu(t)) == cosh(2 * x)
assert _osborne(cosh(x)) == cos(x)
assert _osborne(sinh(x)) == I * sin(x)
assert _osborne(tanh(x)) == I * tan(x)
assert _osborne(coth(x)) == cot(x) / I
assert _osbornei(cos(x)) == cosh(x)
assert _osbornei(sin(x)) == sinh(x) / I
assert _osbornei(tan(x)) == tanh(x) / I
assert _osbornei(cot(x)) == coth(x) * I
assert _osbornei(sec(x)) == 1 / cosh(x)
assert _osbornei(csc(x)) == I / sinh(x)
def test_hyper_as_trig():
from sympy.simplify.fu import _osborne, _osbornei
eq = sinh(x)**2 + cosh(x)**2
t, f = hyper_as_trig(eq)
assert f(fu(t)) == cosh(2*x)
assert _osborne(cosh(x)) == cos(x)
assert _osborne(sinh(x)) == I*sin(x)
assert _osborne(tanh(x)) == I*tan(x)
assert _osborne(coth(x)) == cot(x)/I
assert _osbornei(cos(x)) == cosh(x)
assert _osbornei(sin(x)) == sinh(x)/I
assert _osbornei(tan(x)) == tanh(x)/I
assert _osbornei(cot(x)) == coth(x)*I
assert _osbornei(sec(x)) == 1/cosh(x)
assert _osbornei(csc(x)) == I/sinh(x)
def test_trig_functions(self, printer, x):
# Trig functions
assert printer.doprint(sp.acos(x)) == 'acos(x)'
assert printer.doprint(sp.acosh(x)) == 'acosh(x)'
assert printer.doprint(sp.asin(x)) == 'asin(x)'
assert printer.doprint(sp.asinh(x)) == 'asinh(x)'
assert printer.doprint(sp.atan(x)) == 'atan(x)'
assert printer.doprint(sp.atanh(x)) == 'atanh(x)'
assert printer.doprint(sp.ceiling(x)) == 'ceil(x)'
assert printer.doprint(sp.cos(x)) == 'cos(x)'
assert printer.doprint(sp.cosh(x)) == 'cosh(x)'
assert printer.doprint(sp.exp(x)) == 'exp(x)'
assert printer.doprint(sp.factorial(x)) == 'factorial(x)'
assert printer.doprint(sp.floor(x)) == 'floor(x)'
assert printer.doprint(sp.log(x)) == 'log(x)'
assert printer.doprint(sp.sin(x)) == 'sin(x)'
assert printer.doprint(sp.sinh(x)) == 'sinh(x)'
assert printer.doprint(sp.tan(x)) == 'tan(x)'
assert printer.doprint(sp.tanh(x)) == 'tanh(x)'
# extra trig functions
assert printer.doprint(sp.sec(x)) == '1 / cos(x)'
assert printer.doprint(sp.csc(x)) == '1 / sin(x)'
assert printer.doprint(sp.cot(x)) == '1 / tan(x)'
assert printer.doprint(sp.asec(x)) == 'acos(1 / x)'
assert printer.doprint(sp.acsc(x)) == 'asin(1 / x)'
assert printer.doprint(sp.acot(x)) == 'atan(1 / x)'
assert printer.doprint(sp.sech(x)) == '1 / cosh(x)'
assert printer.doprint(sp.csch(x)) == '1 / sinh(x)'
assert printer.doprint(sp.coth(x)) == '1 / tanh(x)'
assert printer.doprint(sp.asech(x)) == 'acosh(1 / x)'
assert printer.doprint(sp.acsch(x)) == 'asinh(1 / x)'
assert printer.doprint(sp.acoth(x)) == 'atanh(1 / x)'
def test_tan_rewrite():
x = Symbol('x')
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 + S.Pi/2)/cos(x)
assert tan(x).rewrite(cot) == 1/cot(x)
assert tan(sinh(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n()
assert tan(cosh(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n()
assert tan(tanh(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n()
assert tan(coth(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n()
assert tan(sin(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n()
assert tan(cos(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n()
assert tan(tan(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n()
assert tan(cot(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n()
assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
def test_cosh_rewrite():
x = Symbol('x')
assert cosh(x).rewrite(exp) == (exp(x)+exp(-x))/2
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_real_assumptions():
z = Symbol('z', real=False)
assert sinh(z).is_real is None
assert cosh(z).is_real is None
assert tanh(z).is_real is None
assert sech(z).is_real is None
assert csch(z).is_real is None
assert coth(z).is_real is None
def test_sech_rewrite():
x = Symbol("x")
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_real_assumptions():
z = Symbol('z', real=False)
assert sinh(z).is_real is None
assert cosh(z).is_real is None
assert tanh(z).is_real is None
assert sech(z).is_real is None
assert csch(z).is_real is None
assert coth(z).is_real is None
def test_sinh_rewrite():
x = Symbol('x')
assert sinh(x).rewrite(exp) == (exp(x)-exp(-x))/2
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_cosh_rewrite():
x = Symbol('x')
assert cosh(x).rewrite(exp) == (exp(x) + exp(-x)) / 2
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_hyperbolic_simp():
x, y = symbols('x,y')
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) == coth(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) in \
[2 + 3*cosh(x/2)**2, 5 + 3*sinh(x/2)**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), deep=True) == 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
def test_sinh_rewrite():
x = Symbol('x')
assert sinh(x).rewrite(exp) == (exp(x) - exp(-x)) / 2
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():
x = Symbol("x")
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_hyperbolic_simp():
x, y = symbols('x,y')
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) == coth(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) in \
[2 + 3*cosh(x/2)**2, 5 + 3*sinh(x/2)**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), deep=True) == 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
def test_sech_rewrite():
x = Symbol('x')
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_csch_rewrite():
x = Symbol('x')
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_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'
beta = Function('beta')
assert latex(beta(x)) == r"\beta{\left (x \right )}"
assert latex(sin(x)) == r"\sin{\left (x \right )}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
assert latex(asin(x)**2,inv_trig_style="full") == \
r"\arcsin^{2}{\left (x \right )}"
assert latex(asin(x)**2,inv_trig_style="power") == \
r"\sin^{-1}{\left (x \right )}^{2}"
assert latex(asin(x**2),inv_trig_style="power",fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(FallingFactorial(3,
k)) == r"{\left(3\right)}_{\left(k\right)}"
assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"
assert latex(floor(x)) == r"\lfloor{x}\rfloor"
assert latex(ceiling(x)) == r"\lceil{x}\rceil"
assert latex(Abs(x)) == r"\lvert{x}\rvert"
assert latex(re(x)) == r"\Re{x}"
assert latex(re(x + y)) == r"\Re {\left (x + y \right )}"
assert latex(im(x)) == r"\Im{x}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left (x \right )}'
assert latex(coth(x)) == r'\coth{\left (x \right )}'
assert latex(re(x)) == r'\Re{x}'
assert latex(im(x)) == r'\Im{x}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left (x \right )}'
assert latex(zeta(x)) == r'\zeta{\left (x \right )}'
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
def test_complex():
a,b = symbols('a,b', real=True)
z = a + b*I
for func in [sinh, cosh, tanh, coth]:
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)
def test_hyperbolic():
x = Symbol("x")
assert sinh(x).nseries(x, 0, 6) == x + x**3/6 + x**5/120 + O(x**6)
assert cosh(x).nseries(x, 0, 5) == 1 + x**2/2 + x**4/24 + O(x**5)
assert tanh(x).nseries(x, 0, 6) == x - x**3/3 + 2*x**5/15 + O(x**6)
assert coth(x).nseries(x, 0, 6) == 1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**6)
assert asinh(x).nseries(x, 0, 6) == x - x**3/6 + 3*x**5/40 + O(x**6)
assert acosh(x).nseries(x, 0, 6) == pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**6)
assert atanh(x).nseries(x, 0, 6) == x + x**3/3 + x**5/5 + O(x**6)
assert acoth(x).nseries(x, 0, 6) == x + x**3/3 + x**5/5 + pi*I/2 + O(x**6)
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
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1)+exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'
beta = Function('beta')
assert latex(beta(x)) == r"\beta{\left (x \right )}"
assert latex(sin(x)) == r"\sin{\left (x \right )}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
assert latex(asin(x)**2,inv_trig_style="full") == \
r"\arcsin^{2}{\left (x \right )}"
assert latex(asin(x)**2,inv_trig_style="power") == \
r"\sin^{-1}{\left (x \right )}^{2}"
assert latex(asin(x**2),inv_trig_style="power",fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(binomial(2,k)) == r"{\binom{2}{k}}"
assert latex(FallingFactorial(3,k)) == r"{\left(3\right)}_{\left(k\right)}"
assert latex(RisingFactorial(3,k)) == r"{\left(3\right)}^{\left(k\right)}"
assert latex(floor(x)) == r"\lfloor{x}\rfloor"
assert latex(ceiling(x)) == r"\lceil{x}\rceil"
assert latex(Abs(x)) == r"\lvert{x}\rvert"
assert latex(re(x)) == r"\Re{x}"
assert latex(re(x+y)) == r"\Re {\left (x + y \right )}"
assert latex(im(x)) == r"\Im{x}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left (x \right )}'
assert latex(coth(x)) == r'\coth{\left (x \right )}'
assert latex(re(x)) == r'\Re{x}'
assert latex(im(x)) == r'\Im{x}'
assert latex(root(x,y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left (x \right )}'
assert latex(zeta(x)) == r'\zeta{\left (x \right )}'
def test_conjugate():
a = Symbol("a", real=True)
b = Symbol("b", real=True)
x = Symbol("x")
z = a + I * b
zc = a - I * b
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(x)) == sign(x)
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_conv12b():
x = sympy.Symbol("x")
y = sympy.Symbol("y")
assert sympify(sympy.sinh(x/3)) == sinh(Symbol("x") / 3)
assert sympify(sympy.cosh(x/3)) == cosh(Symbol("x") / 3)
assert sympify(sympy.tanh(x/3)) == tanh(Symbol("x") / 3)
assert sympify(sympy.coth(x/3)) == coth(Symbol("x") / 3)
assert sympify(sympy.asinh(x/3)) == asinh(Symbol("x") / 3)
assert sympify(sympy.acosh(x/3)) == acosh(Symbol("x") / 3)
assert sympify(sympy.atanh(x/3)) == atanh(Symbol("x") / 3)
assert sympify(sympy.acoth(x/3)) == acoth(Symbol("x") / 3)
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
def test_conjugate():
a = Symbol("a", real=True)
b = Symbol("b", 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)