def test_series_of_Subs():
from sympy.abc import x, y, z
subs1 = Subs(sin(x), x, y)
subs2 = Subs(sin(x) * cos(z), x, y)
subs3 = Subs(sin(x * z), (x, z), (y, x))
assert subs1.series(x) == subs1
subs1_series = (
Subs(x, x, y) + Subs(-(x ** 3) / 6, x, y) + Subs(x ** 5 / 120, x, y) + O(y ** 6)
)
assert subs1.series() == subs1_series
assert subs1.series(y) == subs1_series
assert subs1.series(z) == subs1
assert subs2.series(z) == (
Subs(z ** 4 * sin(x) / 24, x, y)
+ Subs(-(z ** 2) * sin(x) / 2, x, y)
+ Subs(sin(x), x, y)
+ O(z ** 6)
)
assert subs3.series(x).doit() == subs3.doit().series(x)
assert subs3.series(z).doit() == sin(x * y)
raises(ValueError, lambda: Subs(x + 2 * y, y, z).series())
assert Subs(x + y, y, z).series(x).doit() == x + z
def test_Subs():
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) == \
Subs(f(x, y), (x, y, z), (0, 1, 2))
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x) * y, (x, y), (0, 1)) == Subs(f(y) * x, (y, x), (0, 1))
assert Subs(f(x) * y, (x, y), (1, 1)) == Subs(f(y) * x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y * f(x), x, y).subs(y, 2) == Subs(2 * f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2 * y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2 * f(x), x, 0).diff(y) == 2 * y * f(0)
e = Subs(y**2 * f(x), x, y)
assert e.diff(y) == e.doit().diff(
y) == y**2 * Derivative(f(y), y) + 2 * y * f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2 * Subs(f(x), x, 0)
e1 = Subs(z * f(x), x, 1)
e2 = Subs(z * f(y), y, 1)
assert e1 + e2 == 2 * e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z * f(x + 1), x, 1) not in [e1, e2]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs(x, x + y) == Subs(Derivative(f(x), x),
(x, ), (x + y))
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x * f(x).diff(x).subs(x, 0)).subs(
x, y) == y * f(x).diff(x).subs(x, 0)
def test_Subs():
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) == \
Subs(f(x, y), (x, y, z), (0, 1, 2))
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x)*y, (x, y), (0, 1)) == Subs(f(y)*x, (y, x), (0, 1))
assert Subs(f(x)*y, (x, y), (1, 1)) == Subs(f(y)*x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y*f(x), x, y).subs(y, 2) == Subs(2*f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2*y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2*f(x), x, 0).diff(y) == 2*y*f(0)
e = Subs(y**2*f(x), x, y)
assert e.diff(y) == e.doit().diff(y) == y**2*Derivative(f(y), y) + 2*y*f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2*Subs(f(x), x, 0)
e1 = Subs(z*f(x), x, 1)
e2 = Subs(z*f(y), y, 1)
assert e1 + e2 == 2*e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z*f(x + 1), x, 1) not in [ e1, e2 ]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs(x, x + y) == Subs(Derivative(f(x), x),
(x,), (x + y))
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x*f(x).diff(x).subs(x, 0)).subs(x, y) == y*f(x).diff(x).subs(x, 0)
def test_series_of_Subs():
from sympy.abc import x, y, z
subs1 = Subs(sin(x), x, y)
subs2 = Subs(sin(x) * cos(z), x, y)
subs3 = Subs(sin(x * z), (x, z), (y, x))
assert subs1.series(x) == subs1
assert subs1.series(y) == Subs(x, x, y) + Subs(-x**3/6, x, y) + Subs(x**5/120, x, y) + O(y**6)
assert subs1.series(z) == subs1
assert subs2.series(z) == Subs(z**4*sin(x)/24, x, y) + Subs(-z**2*sin(x)/2, x, y) + Subs(sin(x), x, y) + O(z**6)
assert subs3.series(x).doit() == subs3.doit().series(x)
assert subs3.series(z).doit() == sin(x*y)
def test_Subs():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
f = Function('f')
g = Function('g')
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, 'Subs(f(x, y), (x, y), (0, 0, 1))')
raises(ValueError, 'Subs(f(x, y), (x, x, y), (0, 0, 1))')
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert all([
isinstance(v, Dummy) for v in Subs(f(x, y), (x, y), (0, 1)).variables
])
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x) * y, (x, y), (0, 1)) == Subs(f(y) * x, (y, x), (0, 1))
assert Subs(f(x) * y, (x, y), (1, 1)) == Subs(f(y) * x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1) == Subs(f(x), x, 0)
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y * f(x), x, y).subs(y, 2) == Subs(2 * f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2 * y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == set([y, z])
assert Subs(f(x).diff(x), x, 0).doit() == Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit() == 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2 * f(x), x, 0).diff(y) == 2 * y * f(0)
e = Subs(y**2 * f(x), x, y)
assert e.diff(y) == e.doit().diff(
y) == y**2 * Derivative(f(y), y) + 2 * y * f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2 * Subs(f(x), x, 0)
e1 = Subs(z * f(x), x, 1)
e2 = Subs(z * f(y), y, 1)
assert e1 + e2 == 2 * e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z * f(x + 1), x, 1) not in [e1, e2]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
def test_Subs_subs():
assert Subs(x*y, x, x).subs(x, y) == Subs(x*y, x, y)
assert Subs(x*y, x, x + 1).subs(x, y) == \
Subs(x*y, x, y + 1)
assert Subs(x*y, y, x + 1).subs(x, y) == \
Subs(y**2, y, y + 1)
a = Subs(x*y*z, (y, x, z), (x + 1, x + z, x))
b = Subs(x*y*z, (y, x, z), (x + 1, y + z, y))
assert a.subs(x, y) == b and \
a.doit().subs(x, y) == a.subs(x, y).doit()
f = Function('f')
g = Function('g')
assert Subs(2*f(x, y) + g(x), f(x, y), 1).subs(y, 2) == Subs(
2*f(x, y) + g(x), (f(x, y), y), (1, 2))
def test_Subs_subs():
assert Subs(x*y, x, x).subs(x, y) == Subs(x*y, x, y)
assert Subs(x*y, x, x + 1).subs(x, y) == \
Subs(x*y, x, y + 1)
assert Subs(x*y, y, x + 1).subs(x, y) == \
Subs(y**2, y, y + 1)
a = Subs(x*y*z, (y, x, z), (x + 1, x + z, x))
b = Subs(x*y*z, (y, x, z), (x + 1, y + z, y))
assert a.subs(x, y) == b and \
a.doit().subs(x, y) == a.subs(x, y).doit()
f = Function('f')
g = Function('g')
assert Subs(2*f(x, y) + g(x), f(x, y), 1).subs(y, 2) == Subs(
2*f(x, y) + g(x), (f(x, y), y), (1, 2))
def test_Subs():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
f = Function('f')
g = Function('g')
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, 'Subs(f(x, y), (x, y), (0, 0, 1))')
raises(ValueError, 'Subs(f(x, y), (x, x, y), (0, 0, 1))')
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert all(isinstance(v, Dummy) for v in Subs(f(x, y),
(x, y), (0, 1)).variables)
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x)*y, (x, y), (0, 1)) == Subs(f(y)*x, (y, x), (0, 1))
assert Subs(f(x)*y, (x, y), (1, 1)) == Subs(f(y)*x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1) == Subs(f(x), x, 0)
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y*f(x), x, y).subs(y, 2) == Subs(2*f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2*y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == set([y, z])
assert Subs(f(x).diff(x), x, 0).doit() == Subs(f(x).diff(x), x, 0)
assert Subs(1+f(x).diff(x), x, 0).doit() == 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2*f(x), x, 0).diff(y) == 2*y*f(0)
e = Subs(y**2*f(x), x, y)
assert e.diff(y) == e.doit().diff(y) == y**2*Derivative(f(y), y) + 2*y*f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2*Subs(f(x), x, 0)
e1 = Subs(z*f(x), x, 1)
e2 = Subs(z*f(y), y, 1)
assert e1 + e2 == 2*e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z*f(x+1), x, 1) not in [ e1, e2 ]
assert Derivative(f(x),x).subs(x,g(x)) == Derivative(f(g(x)),g(x))
def test_series_of_Subs():
from sympy.abc import x, y, z
subs1 = Subs(sin(x), x, y)
subs2 = Subs(sin(x) * cos(z), x, y)
subs3 = Subs(sin(x * z), (x, z), (y, x))
assert subs1.series(x) == subs1
assert subs1.series(y) == Subs(x, x, y) + Subs(-x**3 / 6, x, y) + Subs(
x**5 / 120, x, y) + O(y**6)
assert subs1.series(z) == subs1
assert subs2.series(z) == Subs(z**4 * sin(x) / 24, x, y) + Subs(
-z**2 * sin(x) / 2, x, y) + Subs(sin(x), x, y) + O(z**6)
assert subs3.series(x).doit() == subs3.doit().series(x)
assert subs3.series(z).doit() == sin(x * y)
def test_Subs():
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, 'Subs(f(x, y), (x, y), (0, 0, 1))')
raises(ValueError, 'Subs(f(x, y), (x, x, y), (0, 0, 1))')
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert all(
isinstance(v, Dummy) for v in Subs(f(x, y), (x, y), (0, 1)).variables)
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x) * y, (x, y), (0, 1)) == Subs(f(y) * x, (y, x), (0, 1))
assert Subs(f(x) * y, (x, y), (1, 1)) == Subs(f(y) * x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1) == Subs(f(x), x, 0)
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y * f(x), x, y).subs(y, 2) == Subs(2 * f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2 * y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == set([y, z])
assert Subs(f(x).diff(x), x, 0).doit() == Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit() == 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2 * f(x), x, 0).diff(y) == 2 * y * f(0)
e = Subs(y**2 * f(x), x, y)
assert e.diff(y) == e.doit().diff(
y) == y**2 * Derivative(f(y), y) + 2 * y * f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2 * Subs(f(x), x, 0)
e1 = Subs(z * f(x), x, 1)
e2 = Subs(z * f(y), y, 1)
assert e1 + e2 == 2 * e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z * f(x + 1), x, 1) not in [e1, e2]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(1) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(1) == \
z + Rational('1/2').n(1)*f(0)
def test_Subs():
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x ** 2), x ** 2, 0).doit() == f(0)
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, "Subs(f(x, y), (x, y), (0, 0, 1))")
raises(ValueError, "Subs(f(x, y), (x, x, y), (0, 0, 1))")
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert all(isinstance(v, Dummy) for v in Subs(f(x, y), (x, y), (0, 1)).variables)
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x) * y, (x, y), (0, 1)) == Subs(f(y) * x, (y, x), (0, 1))
assert Subs(f(x) * y, (x, y), (1, 1)) == Subs(f(y) * x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1) == Subs(f(x), x, 0)
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y * f(x), x, y).subs(y, 2) == Subs(2 * f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2 * y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == set([y, z])
assert Subs(f(x).diff(x), x, 0).doit() == Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit() == 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y * f(x, y).diff(x), (x, y), (0, 2)).doit() == 2 * Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y ** 2 * f(x), x, 0).diff(y) == 2 * y * f(0)
e = Subs(y ** 2 * f(x), x, y)
assert e.diff(y) == e.doit().diff(y) == y ** 2 * Derivative(f(y), y) + 2 * y * f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2 * Subs(f(x), x, 0)
e1 = Subs(z * f(x), x, 1)
e2 = Subs(z * f(y), y, 1)
assert e1 + e2 == 2 * e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z * f(x + 1), x, 1) not in [e1, e2]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert (
Subs(f(x) * cos(y) + z, (x, y), (0, pi / 3)).n(1)
== Subs(f(x) * cos(y) + z, (x, y), (0, pi / 3)).evalf(1)
== z + Rational("1/2").n(1) * f(0)
)
def test_series_of_Subs():
from sympy.abc import x, y, z
subs1 = Subs(sin(x), x, y)
subs2 = Subs(sin(x) * cos(z), x, y)
subs3 = Subs(sin(x * z), (x, z), (y, x))
assert subs1.series(x) == subs1
subs1_series = (Subs(x, x, y) + Subs(-x**3/6, x, y) +
Subs(x**5/120, x, y) + O(y**6))
assert subs1.series() == subs1_series
assert subs1.series(y) == subs1_series
assert subs1.series(z) == subs1
assert subs2.series(z) == (Subs(z**4*sin(x)/24, x, y) +
Subs(-z**2*sin(x)/2, x, y) + Subs(sin(x), x, y) + O(z**6))
assert subs3.series(x).doit() == subs3.doit().series(x)
assert subs3.series(z).doit() == sin(x*y)
raises(ValueError, lambda: Subs(x + 2*y, y, z).series())
assert Subs(x + y, y, z).series(x).doit() == x + z
def test_Subs():
assert Subs(1, (), ()) is S.One
# check null subs influence on hashing
assert Subs(x, y, z) != Subs(x, y, 1)
# neutral subs works
assert Subs(x, x, 1).subs(x, y).has(y)
# self mapping var/point
assert Subs(Derivative(f(x), (x, 2)), x, x).doit() == f(x).diff(x, x)
assert Subs(x, x, 0).has(x) # it's a structural answer
assert not Subs(x, x, 0).free_symbols
assert Subs(Subs(x + y, x, 2), y, 1) == Subs(x + y, (x, y), (2, 1))
assert Subs(x, (x, ), (0, )) == Subs(x, x, 0)
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(x, y, 2).subs(x, y).doit() == 2
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x) * y, (x, y), (0, 1)) == Subs(f(y) * x, (y, x), (0, 1))
assert Subs(f(x) * y, (x, y), (1, 1)) == Subs(f(y) * x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y * f(x), x, y).subs(y, 2) == Subs(2 * f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2 * y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2 * f(x), x, 0).diff(y) == 2 * y * f(0)
e = Subs(y**2 * f(x), x, y)
assert e.diff(y) == e.doit().diff(
y) == y**2 * Derivative(f(y), y) + 2 * y * f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2 * Subs(f(x), x, 0)
e1 = Subs(z * f(x), x, 1)
e2 = Subs(z * f(y), y, 1)
assert e1 + e2 == 2 * e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z * f(x + 1), x, 1) not in [e1, e2]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs(x, x + y) == Subs(Derivative(f(x), x), x,
x + y)
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x * f(x).diff(x).subs(x, 0)).subs(
x, y) == y * f(x).diff(x).subs(x, 0)
assert Subs(Derivative(g(x)**2, g(x), x), g(x),
exp(x)).doit() == 2 * exp(x)
assert Subs(Derivative(g(x)**2, g(x), x), g(x),
exp(x)).doit(deep=False) == 2 * Derivative(exp(x), x)
assert Derivative(
f(x, g(x)),
x).doit() == Derivative(f(x, g(x)), g(x)) * Derivative(g(x), x) + Subs(
Derivative(f(y, g(x)), y), y, x)
def test_Subs():
assert Subs(1, (), ()) is S.One
# check null subs influence on hashing
assert Subs(x, y, z) != Subs(x, y, 1)
# neutral subs works
assert Subs(x, x, 1).subs(x, y).has(y)
# self mapping var/point
assert Subs(Derivative(f(x), (x, 2)), x, x).doit() == f(x).diff(x, x)
assert Subs(x, x, 0).has(x) # it's a structural answer
assert not Subs(x, x, 0).free_symbols
assert Subs(Subs(x + y, x, 2), y, 1) == Subs(x + y, (x, y), (2, 1))
assert Subs(x, (x,), (0,)) == Subs(x, x, 0)
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(x, y, 2).subs(x, y).doit() == 2
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x)*y, (x, y), (0, 1)) == Subs(f(y)*x, (y, x), (0, 1))
assert Subs(f(x)*y, (x, y), (1, 1)) == Subs(f(y)*x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y*f(x), x, y).subs(y, 2) == Subs(2*f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2*y
assert Subs(f(x), x, 0).free_symbols == set([])
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2*f(x), x, 0).diff(y) == 2*y*f(0)
e = Subs(y**2*f(x), x, y)
assert e.diff(y) == e.doit().diff(y) == y**2*Derivative(f(y), y) + 2*y*f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2*Subs(f(x), x, 0)
e1 = Subs(z*f(x), x, 1)
e2 = Subs(z*f(y), y, 1)
assert e1 + e2 == 2*e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z*f(x + 1), x, 1) not in [ e1, e2 ]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs(x, x + y) == Subs(Derivative(f(x), x),
x, x + y)
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x*f(x).diff(x).subs(x, 0)).subs(x, y) == y*f(x).diff(x).subs(x, 0)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit() == 2*exp(x)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit(deep=False) == 2*Derivative(exp(x), x)
assert Derivative(f(x, g(x)), x).doit() == Derivative(g(x), x
)*Subs(Derivative(f(x, y), y), y, g(x)
) + Subs(Derivative(f(y, g(x)), y), y, x)