def test_issue_5223():
assert series(1, x) == 1
assert next(S.Zero.lseries(x)) == 0
assert cos(x).series() == cos(x).series(x)
raises(ValueError, lambda: cos(x + y).series())
raises(ValueError, lambda: x.series(dir=""))
assert (cos(x).series(x, 1) - cos(x + 1).series(x).subs(x, x - 1)).removeO() == 0
e = cos(x).series(x, 1, n=None)
assert [next(e) for i in range(2)] == [cos(1), -((x - 1) * sin(1))]
e = cos(x).series(x, 1, n=None, dir="-")
assert [next(e) for i in range(2)] == [cos(1), (1 - x) * sin(1)]
# the following test is exact so no need for x -> x - 1 replacement
assert abs(x).series(x, 1, dir="-") == x
assert (
exp(x).series(x, 1, dir="-", n=3).removeO()
== E - E * (-x + 1) + E * (-x + 1) ** 2 / 2
)
D = Derivative
assert D(x ** 2 + x ** 3 * y ** 2, x, 2, y, 1).series(x).doit() == 12 * x * y
assert next(D(cos(x), x).lseries()) == D(1, x)
assert D(exp(x), x).series(n=3) == D(1, x) + D(x, x) + D(x ** 2 / 2, x) + D(
x ** 3 / 6, x
) + O(x ** 3)
assert Integral(x, (x, 1, 3), (y, 1, x)).series(x) == -4 + 4 * x
assert (1 + x + O(x ** 2)).getn() == 2
assert (1 + x).getn() is None
assert ((1 / sin(x)) ** oo).series() is oo
logx = Symbol("logx")
assert ((sin(x)) ** y).nseries(x, n=1, logx=logx) == exp(y * logx) + O(
x * exp(y * logx), x
)
assert sin(1 / x).series(x, oo, n=5) == 1 / x - 1 / (6 * x ** 3) + O(
x ** (-5), (x, oo)
)
assert abs(x).series(x, oo, n=5, dir="+") == x
assert abs(x).series(x, -oo, n=5, dir="-") == -x
assert abs(-x).series(x, oo, n=5, dir="+") == x
assert abs(-x).series(x, -oo, n=5, dir="-") == -x
assert exp(x * log(x)).series(n=3) == 1 + x * log(x) + x ** 2 * log(x) ** 2 / 2 + O(
x ** 3 * log(x) ** 3
)
# XXX is this right? If not, fix "ngot > n" handling in expr.
p = Symbol("p", positive=True)
assert exp(sqrt(p) ** 3 * log(p)).series(n=3) == 1 + p ** S("3/2") * log(p) + O(
p ** 3 * log(p) ** 3
)
assert exp(sin(x) * log(x)).series(n=2) == 1 + x * log(x) + O(x ** 2 * log(x) ** 2)
def test_2124():
assert series(1, x) == 1
assert S(0).lseries(x).next() == 0
assert cos(x).series() == cos(x).series(x)
raises(ValueError, lambda: cos(x + y).series())
raises(ValueError, lambda: x.series(dir=""))
assert (cos(x).series(x, 1).removeO().subs(x, x - 1) -
cos(x + 1).series(x).removeO().subs(x, x - 1)).expand() == 0
e = cos(x).series(x, 1, n=None)
assert [e.next() for i in range(2)] == [cos(1), -((x - 1) * sin(1))]
e = cos(x).series(x, 1, n=None, dir='-')
assert [e.next() for i in range(2)] == [cos(1), (1 - x) * sin(1)]
# the following test is exact so no need for x -> x - 1 replacement
assert abs(x).series(x, 1, dir='-') == x
assert exp(x).series(x, 1, dir='-', n=3).removeO().subs(x, x - 1) == \
E + E*(x - 1) + E*(x - 1)**2/2
D = Derivative
assert D(x**2 + x**3 * y**2, x, 2, y, 1).series(x).doit() == 12 * x * y
assert D(cos(x), x).lseries().next() == D(1, x)
assert D(exp(x),
x).series(n=3) == D(1, x) + D(x, x) + D(x**2 / 2, x) + O(x**3)
assert Integral(x, (x, 1, 3), (y, 1, x)).series(x) == -4 + 4 * x
assert (1 + x + O(x**2)).getn() == 2
assert (1 + x).getn() is None
assert ((1 / sin(x))**oo).series() == oo
logx = Symbol('logx')
assert ((sin(x))**y).nseries(x, n=1, logx=logx) == \
exp(y*logx) + O(x*exp(y*logx), x)
raises(NotImplementedError, lambda: series(Function("f")(x)))
assert sin(1 / x).series(x, oo, n=5) == 1 / x - 1 / (6 * x**3)
assert abs(x).series(x, oo, n=5, dir='+') == x
assert abs(x).series(x, -oo, n=5, dir='-') == -x
assert abs(-x).series(x, oo, n=5, dir='+') == x
assert abs(-x).series(x, -oo, n=5, dir='-') == -x
assert exp(x*log(x)).series(n=3) == \
1 + x*log(x) + x**2*log(x)**2/2 + O(x**3*log(x)**3)
# XXX is this right? If not, fix "ngot > n" handling in expr.
p = Symbol('p', positive=True)
assert exp(sqrt(p)**3*log(p)).series(n=3) == \
1 + p**S('3/2')*log(p) + O(p**3*log(p)**3)
assert exp(sin(x) *
log(x)).series(n=2) == 1 + x * log(x) + O(x**2 * log(x)**2)
def test_2124():
assert series(1, x) == 1
assert S(0).lseries(x).next() == 0
assert cos(x).series() == cos(x).series(x)
raises(ValueError, lambda: cos(x + y).series())
raises(ValueError, lambda: x.series(dir=""))
assert (cos(x).series(x, 1).removeO().subs(x, x - 1) -
cos(x + 1).series(x).removeO().subs(x, x - 1)).expand() == 0
e = cos(x).series(x, 1, n=None)
assert [e.next() for i in range(2)] == [cos(1), -((x - 1)*sin(1))]
e = cos(x).series(x, 1, n=None, dir='-')
assert [e.next() for i in range(2)] == [cos(1), (1 - x)*sin(1)]
# the following test is exact so no need for x -> x - 1 replacement
assert abs(x).series(x, 1, dir='-') == x
assert exp(x).series(x, 1, dir='-', n=3).removeO().subs(x, x - 1) == \
E + E*(x - 1) + E*(x - 1)**2/2
D = Derivative
assert D(x**2 + x**3*y**2, x, 2, y, 1).series(x).doit() == 12*x*y
assert D(cos(x), x).lseries().next() == D(1, x)
assert D(
exp(x), x).series(n=3) == D(1, x) + D(x, x) + D(x**2/2, x) + O(x**3)
assert Integral(x, (x, 1, 3), (y, 1, x)).series(x) == -4 + 4*x
assert (1 + x + O(x**2)).getn() == 2
assert (1 + x).getn() is None
assert ((1/sin(x))**oo).series() == oo
logx = Symbol('logx')
assert ((sin(x))**y).nseries(x, n=1, logx=logx) \
== exp(y*logx) + O(x*exp(y*logx), x)
raises(NotImplementedError, lambda: series(Function("f")(x)))
assert sin(1/x).series(x, oo, n=5) == 1/x - 1/(6*x**3)
assert abs(x).series(x, oo, n=5, dir='+') == x
assert abs(x).series(x, -oo, n=5, dir='-') == -x
assert abs(-x).series(x, oo, n=5, dir='+') == x
assert abs(-x).series(x, -oo, n=5, dir='-') == -x
assert exp(x*log(x)).series(n=3) == \
1 + x*log(x) + x**2*log(x)**2/2 + O(x**3*log(x)**3)
# XXX is this right? If not, fix "ngot > n" handling in expr.
p = Symbol('p', positive=True)
assert exp(sqrt(p)**3*log(p)).series(n=3) == \
1 + p**S('3/2')*log(p) + O(p**3*log(p)**3)
assert exp(sin(x)*log(x)).series(n=2) == 1 + x*log(x) + O(x**2*log(x)**2)
