def test_diff():
x, y = symbols('x, y')
assert Rational(1, 3).diff(x) is S.Zero
assert I.diff(x) is S.Zero
assert pi.diff(x) is S.Zero
assert x.diff(x, 0) == x
assert (x**2).diff(x, 2, x) == 0
assert (x**2).diff(x, y, 0) == 2 * x
assert (x**2).diff(x, y) == 0
raises(ValueError, lambda: x.diff(1, x))
a = Symbol("a")
b = Symbol("b")
c = Symbol("c")
p = Rational(5)
e = a * b + b**p
assert e.diff(a) == b
assert e.diff(b) == a + 5 * b**4
assert e.diff(b).diff(a) == Rational(1)
e = a * (b + c)
assert e.diff(a) == b + c
assert e.diff(b) == a
assert e.diff(b).diff(a) == Rational(1)
e = c**p
assert e.diff(c, 6) == Rational(0)
assert e.diff(c, 5) == Rational(120)
e = c**Rational(2)
assert e.diff(c) == 2 * c
e = a * b * c
assert e.diff(c) == a * b
def test_diff():
x, y = symbols('x, y')
assert Rational(1, 3).diff(x) is S.Zero
assert I.diff(x) is S.Zero
assert pi.diff(x) is S.Zero
assert x.diff(x, 0) == x
assert (x**2).diff(x, 2, x) == 0
assert (x**2).diff(x, y, 0) == 2*x
assert (x**2).diff(x, y) == 0
raises(ValueError, lambda: x.diff(1, x))
a = Symbol("a")
b = Symbol("b")
c = Symbol("c")
p = Rational(5)
e = a*b + b**p
assert e.diff(a) == b
assert e.diff(b) == a + 5*b**4
assert e.diff(b).diff(a) == Rational(1)
e = a*(b + c)
assert e.diff(a) == b + c
assert e.diff(b) == a
assert e.diff(b).diff(a) == Rational(1)
e = c**p
assert e.diff(c, 6) == Rational(0)
assert e.diff(c, 5) == Rational(120)
e = c**Rational(2)
assert e.diff(c) == 2*c
e = a*b*c
assert e.diff(c) == a*b
def reduce_rational_inequalities_wrap(condition, var):
if condition.is_Relational:
return _reduce_inequalities([[condition]], var, relational=False)
if isinstance(condition, Or):
return Union(*[_reduce_inequalities([[arg]], var, relational=False)
for arg in condition.args])
if isinstance(condition, And):
intervals = [_reduce_inequalities([[arg]], var, relational=False)
for arg in condition.args]
I = intervals[0]
for i in intervals:
I = I.intersect(i)
return I
def reduce_rational_inequalities_wrap(condition, var):
if condition.is_Relational:
return _reduce_inequalities([[condition]], var, relational=False)
if condition.__class__ is Or:
return _reduce_inequalities([list(condition.args)],
var, relational=False)
if condition.__class__ is And:
intervals = [_reduce_inequalities([[arg]], var, relational=False)
for arg in condition.args]
I = intervals[0]
for i in intervals:
I = I.intersect(i)
return I
def test_issue_8919():
c = symbols('c:5')
x = symbols("x")
f1 = Piecewise((c[1], x < 1), (c[2], True))
f2 = Piecewise((c[3], x < S(1) / 3), (c[4], True))
assert integrate(
f1 * f2,
(x, 0, 2)) == c[1] * c[3] / 3 + 2 * c[1] * c[4] / 3 + c[2] * c[4]
f1 = Piecewise((0, x < 1), (2, True))
f2 = Piecewise((3, x < 2), (0, True))
assert integrate(f1 * f2, (x, 0, 3)) == 6
y = symbols("y", positive=True)
a, b, c, x, z = symbols("a,b,c,x,z", real=True)
I = Integral(Piecewise((0, (x >= y) | (x < 0) | (b > c)), (a, True)),
(x, 0, z))
ans = I.doit()
assert ans == Piecewise((0, b > c), (a * z - a * Min(0, z), True))
for cond in (True, False):
for yy in range(1, 3):
for zz in range(-1, 0, 1):
reps = [(b > c, cond), (y, yy), (z, zz)]
assert ans.subs(reps) == I.subs(reps).doit()
def test_issue_8919():
c = symbols('c:5')
x = symbols("x")
f1 = Piecewise((c[1], x < 1), (c[2], True))
f2 = Piecewise((c[3], x < S(1)/3), (c[4], True))
assert integrate(f1*f2, (x, 0, 2)
) == c[1]*c[3]/3 + 2*c[1]*c[4]/3 + c[2]*c[4]
f1 = Piecewise((0, x < 1), (2, True))
f2 = Piecewise((3, x < 2), (0, True))
assert integrate(f1*f2, (x, 0, 3)) == 6
y = symbols("y", positive=True)
a, b, c, x, z = symbols("a,b,c,x,z", real=True)
I = Integral(Piecewise(
(0, (x >= y) | (x < 0) | (b > c)),
(a, True)), (x, 0, z))
ans = I.doit()
assert ans == Piecewise((0, b > c), (a*Min(y, z) - a*Min(0, z), True))
for cond in (True, False):
for yy in range(1, 3):
for zz in range(-yy, 0, yy):
reps = [(b > c, cond), (y, yy), (z, zz)]
assert ans.subs(reps) == I.subs(reps).doit()
def test_issue_2387_bug():
from sympy import I, Expr
assert abs(Expr._from_mpmath(I._to_mpmath(15), 15) - I) < 1.0e-15
def test_issue_21721():
a = Symbol('a', real=True)
I = integrate(1/(pi*(1 + (x - a)**2)), x)
assert I.limit(x, oo) == S.Half
