def test_issue_11730():
unit = Interval(0, 1)
square = ComplexRegion(unit**2)
assert Union(S.Complexes, FiniteSet(oo)) != S.Complexes
assert Union(S.Complexes, FiniteSet(eye(4))) != S.Complexes
assert Union(unit, square) == square
assert Intersection(S.Reals, square) == unit
def test_issue_11938():
unit = Interval(0, 1)
ival = Interval(1, 2)
cr1 = ComplexRegion(ival * unit)
assert Intersection(cr1, S.Reals) == ival
assert Intersection(cr1, unit) == FiniteSet(1)
arg1 = Interval(0, S.Pi)
arg2 = FiniteSet(S.Pi)
arg3 = Interval(S.Pi / 4, 3 * S.Pi / 4)
cp1 = ComplexRegion(unit * arg1, polar=True)
cp2 = ComplexRegion(unit * arg2, polar=True)
cp3 = ComplexRegion(unit * arg3, polar=True)
assert Intersection(cp1, S.Reals) == Interval(-1, 1)
assert Intersection(cp2, S.Reals) == Interval(-1, 0)
assert Intersection(cp3, S.Reals) == FiniteSet(0)
def _(self, other):
if other.is_ComplexRegion:
# self in rectangular form
if (not self.polar) and (not other.polar):
return ComplexRegion(Intersection(self.sets, other.sets))
# self in polar form
elif self.polar and other.polar:
r1, theta1 = self.a_interval, self.b_interval
r2, theta2 = other.a_interval, other.b_interval
new_r_interval = Intersection(r1, r2)
new_theta_interval = Intersection(theta1, theta2)
# 0 and 2*Pi means the same
if ((2 * S.Pi in theta1 and S.Zero in theta2)
or (2 * S.Pi in theta2 and S.Zero in theta1)):
new_theta_interval = Union(new_theta_interval, FiniteSet(0))
return ComplexRegion(new_r_interval * new_theta_interval,
polar=True)
if other.is_subset(S.Reals):
new_interval = []
x = symbols("x", cls=Dummy, real=True)
# self in rectangular form
if not self.polar:
for element in self.psets:
if S.Zero in element.args[1]:
new_interval.append(element.args[0])
new_interval = Union(*new_interval)
return Intersection(new_interval, other)
# self in polar form
elif self.polar:
for element in self.psets:
if S.Zero in element.args[1]:
new_interval.append(element.args[0])
if S.Pi in element.args[1]:
new_interval.append(
ImageSet(Lambda(x, -x), element.args[0]))
if S.Zero in element.args[0]:
new_interval.append(FiniteSet(0))
new_interval = Union(*new_interval)
return Intersection(new_interval, other)
def test_Complex():
assert 5 in S.Complexes
assert 5 + 4 * I in S.Complexes
assert S.Pi in S.Complexes
assert -sqrt(2) in S.Complexes
assert -I in S.Complexes
assert sqrt(-1) in S.Complexes
assert S.Complexes.intersect(S.Reals) == S.Reals
# assert S.Complexes.union(S.Reals) == S.Complexes
assert S.Complexes == ComplexRegion(S.Reals * S.Reals)
def test_ImageSet():
raises(ValueError, lambda: ImageSet(x, S.Integers))
assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1)
assert ImageSet(Lambda(x, y), S.Integers) == {y}
assert ImageSet(Lambda(x, 1), S.EmptySet) == S.EmptySet
empty = Intersection(FiniteSet(log(2) / pi), S.Integers)
assert unchanged(ImageSet, Lambda(x, 1), empty) # issue #17471
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1 / x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert (1, 2) not in harmonics
assert harmonics.is_iterable
assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0)
assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4)
c = ComplexRegion(Interval(1, 3) * Interval(1, 3))
assert Tuple(2, 6) in ImageSet(Lambda((x, y), (x, 2 * y)), c)
assert Tuple(2, S.Half) in ImageSet(Lambda((x, y), (x, 1 / y)), c)
assert Tuple(2, -2) not in ImageSet(Lambda((x, y), (x, y**2)), c)
assert Tuple(2, -2) in ImageSet(Lambda((x, y), (x, -2)), c)
c3 = Interval(3, 7) * Interval(8, 11) * Interval(5, 9)
assert Tuple(8, 3, 9) in ImageSet(Lambda((t, y, x), (y, t, x)), c3)
assert Tuple(Rational(1, 8), 3,
9) in ImageSet(Lambda((t, y, x), (1 / y, t, x)), c3)
assert 2 / pi not in ImageSet(Lambda((x, y), 2 / x), c)
assert 2 / S(100) not in ImageSet(Lambda((x, y), 2 / x), c)
assert Rational(2, 3) in ImageSet(Lambda((x, y), 2 / x), c)
assert imageset(lambda x, y: x + y, S.Integers,
S.Naturals).base_set == ProductSet(S.Integers, S.Naturals)
# Passing a set instead of a FiniteSet shouldn't raise
assert unchanged(ImageSet, Lambda(x, x**2), {1, 2, 3})
raises(TypeError, lambda: ImageSet(Lambda(x, x**2), 1))
def test_ComplexRegion_contains():
# contains in ComplexRegion
a = Interval(2, 3)
b = Interval(4, 6)
c = Interval(7, 9)
c1 = ComplexRegion(a * b)
c2 = ComplexRegion(Union(a * b, c * a))
assert 2.5 + 4.5 * I in c1
assert 2 + 4 * I in c1
assert 3 + 4 * I in c1
assert 8 + 2.5 * I in c2
assert 2.5 + 6.1 * I not in c1
assert 4.5 + 3.2 * I not in c1
r1 = Interval(0, 1)
theta1 = Interval(0, 2 * S.Pi)
c3 = ComplexRegion(r1 * theta1, polar=True)
assert 0.5 + 0.6 * I in c3
assert I in c3
assert 1 in c3
assert 0 in c3
assert 1 + I not in c3
assert 1 - I not in c3
def test_ImageSet():
raises(ValueError, lambda: ImageSet(x, S.Integers))
assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1)
assert ImageSet(Lambda(x, y), S.Integers) == {y}
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1 / x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert (1, 2) not in harmonics
assert harmonics.is_iterable
assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0)
assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4)
c = ComplexRegion(Interval(1, 3) * Interval(1, 3))
assert Tuple(2, 6) in ImageSet(Lambda((x, y), (x, 2 * y)), c)
assert Tuple(2, S.Half) in ImageSet(Lambda((x, y), (x, 1 / y)), c)
assert Tuple(2, -2) not in ImageSet(Lambda((x, y), (x, y**2)), c)
assert Tuple(2, -2) in ImageSet(Lambda((x, y), (x, -2)), c)
c3 = Interval(3, 7) * Interval(8, 11) * Interval(5, 9)
assert Tuple(8, 3, 9) in ImageSet(Lambda((t, y, x), (y, t, x)), c3)
assert Tuple(S(1) / 8, 3, 9) in ImageSet(Lambda((t, y, x), (1 / y, t, x)),
c3)
assert 2 / pi not in ImageSet(Lambda((x, y), 2 / x), c)
assert 2 / S(100) not in ImageSet(Lambda((x, y), 2 / x), c)
assert 2 / S(3) in ImageSet(Lambda((x, y), 2 / x), c)
assert imageset(lambda x, y: x + y, S.Integers,
S.Naturals).base_set == ProductSet(S.Integers, S.Naturals)
def test_ComplexRegion_union():
# Polar form
c1 = ComplexRegion(Interval(0, 1) * Interval(0, 2 * S.Pi), polar=True)
c2 = ComplexRegion(Interval(0, 1) * Interval(0, S.Pi), polar=True)
c3 = ComplexRegion(Interval(0, oo) * Interval(0, S.Pi), polar=True)
c4 = ComplexRegion(Interval(0, oo) * Interval(S.Pi, 2 * S.Pi), polar=True)
p1 = Union(
Interval(0, 1) * Interval(0, 2 * S.Pi),
Interval(0, 1) * Interval(0, S.Pi))
p2 = Union(
Interval(0, oo) * Interval(0, S.Pi),
Interval(0, oo) * Interval(S.Pi, 2 * S.Pi))
assert c1.union(c2) == ComplexRegion(p1, polar=True)
assert c3.union(c4) == ComplexRegion(p2, polar=True)
# Rectangular form
c5 = ComplexRegion(Interval(2, 5) * Interval(6, 9))
c6 = ComplexRegion(Interval(4, 6) * Interval(10, 12))
c7 = ComplexRegion(Interval(0, 10) * Interval(-10, 0))
c8 = ComplexRegion(Interval(12, 16) * Interval(14, 20))
p3 = Union(
Interval(2, 5) * Interval(6, 9),
Interval(4, 6) * Interval(10, 12))
p4 = Union(
Interval(0, 10) * Interval(-10, 0),
Interval(12, 16) * Interval(14, 20))
assert c5.union(c6) == ComplexRegion(p3)
assert c7.union(c8) == ComplexRegion(p4)
assert c1.union(Interval(2, 4)) == Union(c1,
Interval(2, 4),
evaluate=False)
assert c5.union(Interval(2, 4)) == Union(c5,
Interval(2, 4),
evaluate=False)
def test_ComplexRegion_intersect():
# Polar form
X_axis = ComplexRegion(Interval(0, oo) * FiniteSet(0, S.Pi), polar=True)
unit_disk = ComplexRegion(Interval(0, 1) * Interval(0, 2 * S.Pi),
polar=True)
upper_half_unit_disk = ComplexRegion(Interval(0, 1) * Interval(0, S.Pi),
polar=True)
upper_half_disk = ComplexRegion(Interval(0, oo) * Interval(0, S.Pi),
polar=True)
lower_half_disk = ComplexRegion(Interval(0, oo) * Interval(S.Pi, 2 * S.Pi),
polar=True)
right_half_disk = ComplexRegion(Interval(0, oo) *
Interval(-S.Pi / 2, S.Pi / 2),
polar=True)
first_quad_disk = ComplexRegion(Interval(0, oo) * Interval(0, S.Pi / 2),
polar=True)
assert upper_half_disk.intersect(unit_disk) == upper_half_unit_disk
assert right_half_disk.intersect(first_quad_disk) == first_quad_disk
assert upper_half_disk.intersect(right_half_disk) == first_quad_disk
assert upper_half_disk.intersect(lower_half_disk) == X_axis
c1 = ComplexRegion(Interval(0, 4) * Interval(0, 2 * S.Pi), polar=True)
assert c1.intersect(Interval(1, 5)) == Interval(1, 4)
assert c1.intersect(Interval(4, 9)) == FiniteSet(4)
assert c1.intersect(Interval(5, 12)) is S.EmptySet
# Rectangular form
X_axis = ComplexRegion(Interval(-oo, oo) * FiniteSet(0))
unit_square = ComplexRegion(Interval(-1, 1) * Interval(-1, 1))
upper_half_unit_square = ComplexRegion(Interval(-1, 1) * Interval(0, 1))
upper_half_plane = ComplexRegion(Interval(-oo, oo) * Interval(0, oo))
lower_half_plane = ComplexRegion(Interval(-oo, oo) * Interval(-oo, 0))
right_half_plane = ComplexRegion(Interval(0, oo) * Interval(-oo, oo))
first_quad_plane = ComplexRegion(Interval(0, oo) * Interval(0, oo))
assert upper_half_plane.intersect(unit_square) == upper_half_unit_square
assert right_half_plane.intersect(first_quad_plane) == first_quad_plane
assert upper_half_plane.intersect(right_half_plane) == first_quad_plane
assert upper_half_plane.intersect(lower_half_plane) == X_axis
c1 = ComplexRegion(Interval(-5, 5) * Interval(-10, 10))
assert c1.intersect(Interval(2, 7)) == Interval(2, 5)
assert c1.intersect(Interval(5, 7)) == FiniteSet(5)
assert c1.intersect(Interval(6, 9)) is S.EmptySet
# unevaluated object
C1 = ComplexRegion(Interval(0, 1) * Interval(0, 2 * S.Pi), polar=True)
C2 = ComplexRegion(Interval(-1, 1) * Interval(-1, 1))
assert C1.intersect(C2) == Intersection(C1, C2, evaluate=False)
def test_ComplexRegion_from_real():
c1 = ComplexRegion(Interval(0, 1) * Interval(0, 2 * S.Pi), polar=True)
raises(ValueError, lambda: c1.from_real(c1))
assert c1.from_real(Interval(-1, 1)) == ComplexRegion(
Interval(-1, 1) * FiniteSet(0), False)
