def test_Lambda():
e = Lambda(x, x**2)
assert e(4) == 16
assert e(x) == x**2
assert e(y) == y**2
assert Lambda((), 42)() == 42
assert Lambda((), 42) == Lambda((), 42)
assert Lambda((), 42) != Lambda((), 43)
assert Lambda((), f(x))() == f(x)
assert Lambda((), 42).nargs == FiniteSet(0)
assert Lambda(x, x**2) == Lambda(x, x**2)
assert Lambda(x, x**2) == Lambda(y, y**2)
assert Lambda(x, x**2) != Lambda(y, y**2 + 1)
assert Lambda((x, y), x**y) == Lambda((y, x), y**x)
assert Lambda((x, y), x**y) != Lambda((x, y), y**x)
assert Lambda((x, y), x**y)(x, y) == x**y
assert Lambda((x, y), x**y)(3, 3) == 3**3
assert Lambda((x, y), x**y)(x, 3) == x**3
assert Lambda((x, y), x**y)(3, y) == 3**y
assert Lambda(x, f(x))(x) == f(x)
assert Lambda(x, x**2)(e(x)) == x**4
assert e(e(x)) == x**4
assert Lambda((x, y), x + y).nargs == FiniteSet(2)
p = x, y, z, t
assert Lambda(p, t*(x + y + z))(*p) == t * (x + y + z)
assert Lambda(x, 2*x) + Lambda(y, 2*y) == 2*Lambda(x, 2*x)
assert Lambda(x, 2*x) not in [ Lambda(x, x) ]
raises(TypeError, lambda: Lambda(1, x))
assert Lambda(x, 1)(1) is S.One
def test_f_expand_complex():
x = Symbol("x", real=True)
assert f(x).expand(complex=True) == I * im(f(x)) + re(f(x))
assert exp(x).expand(complex=True) == exp(x)
assert exp(I * x).expand(complex=True) == cos(x) + I * sin(x)
assert exp(z).expand(complex=True) == cos(im(z)) * exp(re(z)) + I * sin(im(z)) * exp(re(z))
def test_derivative_subs_bug():
e = diff(g(x), x)
assert e.subs(g(x), f(x)) != e
assert e.subs(g(x), f(x)) == Derivative(f(x), x)
assert e.subs(g(x), -f(x)) == Derivative(-f(x), x)
assert e.subs(x, y) == Derivative(g(y), y)
def test_diff():
x = Symbol("x")
y = Symbol("y")
f = Function("f")
g = Function("g")
assert simplify(g(x).diff(x)*f(x).diff(x) - f(x).diff(x)*g(x).diff(x)) == 0
assert simplify(2*f(x)*f(x).diff(x) - diff(f(x)**2, x)) == 0
assert simplify(diff(1/f(x), x) + f(x).diff(x)/f(x)**2) == 0
assert simplify(f(x).diff(x, y) - f(x).diff(y, x)) == 0
def test_is_commutative():
from sympy.physics.secondquant import NO, F, Fd
m = Symbol('m', commutative=False)
for f in (Sum, Product, Integral):
assert f(z, (z, 1, 1)).is_commutative is True
assert f(z*y, (z, 1, 6)).is_commutative is True
assert f(m*x, (x, 1, 2)).is_commutative is False
assert f(NO(Fd(x)*F(y))*z, (z, 1, 2)).is_commutative is False
def test_function_complex():
x = Symbol('x', complex=True)
assert f(x).is_commutative is True
assert sin(x).is_commutative is True
assert exp(x).is_commutative is True
assert log(x).is_commutative is True
assert f(x).is_complex is True
assert sin(x).is_complex is True
assert exp(x).is_complex is True
assert log(x).is_complex is True
def test_function_non_commutative():
x = Symbol('x', commutative=False)
assert f(x).is_commutative is False
assert sin(x).is_commutative is False
assert exp(x).is_commutative is False
assert log(x).is_commutative is False
assert f(x).is_complex is False
assert sin(x).is_complex is False
assert exp(x).is_complex is False
assert log(x).is_complex is False
def test_unhandled():
class MyExpr(Expr):
def _eval_derivative(self, s):
if not s.name.startswith('xi'):
return self
else:
return None
expr = MyExpr(x, y, z)
assert diff(expr, x, y, f(x), z) == Derivative(expr, f(x), z)
assert diff(expr, f(x), x) == Derivative(expr, f(x), x)
def test_simplify_expr():
x, y, z, k, n, m, w, f, s, A = symbols("x,y,z,k,n,m,w,f,s,A")
assert all(simplify(tmp) == tmp for tmp in [I, E, oo, x, -x, -oo, -E, -I])
e = 1 / x + 1 / y
assert e != (x + y) / (x * y)
assert simplify(e) == (x + y) / (x * y)
e = A ** 2 * s ** 4 / (4 * pi * k * m ** 3)
assert simplify(e) == e
e = (4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)
assert simplify(e) == 0
e = (-4 * x * y ** 2 - 2 * y ** 3 - 2 * x ** 2 * y) / (x + y) ** 2
assert simplify(e) == -2 * y
e = -x - y - (x + y) ** (-1) * y ** 2 + (x + y) ** (-1) * x ** 2
assert simplify(e) == -2 * y
e = (x + x * y) / x
assert simplify(e) == 1 + y
e = (f(x) + y * f(x)) / f(x)
assert simplify(e) == 1 + y
e = (2 * (1 / n - cos(n * pi) / n)) / pi
assert simplify(e) == (-cos(pi * n) + 1) / (pi * n) * 2
e = integrate(1 / (x ** 3 + 1), x).diff(x)
assert simplify(e) == 1 / (x ** 3 + 1)
e = integrate(x / (x ** 2 + 3 * x + 1), x).diff(x)
assert simplify(e) == x / (x ** 2 + 3 * x + 1)
A = Matrix([[2 * k - m * w ** 2, -k], [-k, k - m * w ** 2]]).inv()
assert simplify((A * Matrix([0, f]))[1]) == -f * (2 * k - m * w ** 2) / (
k ** 2 - (k - m * w ** 2) * (2 * k - m * w ** 2)
)
f = -x + y / (z + t) + z * x / (z + t) + z * a / (z + t) + t * x / (z + t)
assert simplify(f) == (y + a * z) / (z + t)
A, B = symbols("A,B", commutative=False)
assert simplify(A * B - B * A) == A * B - B * A
assert simplify(A / (1 + y / x)) == x * A / (x + y)
assert simplify(A * (1 / x + 1 / y)) == A / x + A / y # (x + y)*A/(x*y)
assert simplify(log(2) + log(3)) == log(6)
assert simplify(log(2 * x) - log(2)) == log(x)
assert simplify(hyper([], [], x)) == exp(x)
def test_simplify_function_inverse():
x, y = symbols('x, y')
g = Function('g')
class f(Function):
def inverse(self, argindex=1):
return g
assert simplify(f(g(x))) == x
assert simplify(f(g(sin(x)**2 + cos(x)**2))) == 1
assert simplify(f(g(x, y))) == f(g(x, y))
def test_issue_7068():
from sympy.abc import a, b, f
y1 = Dummy('y')
y2 = Dummy('y')
func1 = f(a + y1 * b)
func2 = f(a + y2 * b)
func1_y = func1.diff(y1)
func2_y = func2.diff(y2)
assert func1_y != func2_y
z1 = Subs(f(a), a, y1)
z2 = Subs(f(a), a, y2)
assert z1 != z2
def test_issue_12005():
e1 = Subs(Derivative(f(x), x), (x,), (x,))
assert e1.diff(x) == Derivative(f(x), x, x)
e2 = Subs(Derivative(f(x), x), (x,), (x**2 + 1,))
assert e2.diff(x) == 2*x*Subs(Derivative(f(x), x, x), (x,), (x**2 + 1,))
e3 = Subs(Derivative(f(x) + y**2 - y, y), (y,), (y**2,))
assert e3.diff(y) == 4*y
e4 = Subs(Derivative(f(x + y), y), (y,), (x**2))
assert e4.diff(y) == S.Zero
e5 = Subs(Derivative(f(x), x), (y, z), (y, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Derivative(f(g(x)), g(x), g(x))
def test_func_deriv():
assert f(x).diff(x) == Derivative(f(x), x)
# issue 4534
assert f(x, y).diff(x, y) - f(x, y).diff(y, x) == 0
assert Derivative(f(x, y), x, y).args[1:] == ((x, 1), (y, 1))
assert Derivative(f(x, y), y, x).args[1:] == ((y, 1), (x, 1))
assert (Derivative(f(x, y), x, y) - Derivative(f(x, y), y, x)).doit() == 0
def test_Sum_doit():
assert Sum(n*Integral(a**2), (n, 0, 2)).doit() == a**3
assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep=False) == \
3*Integral(a**2)
assert summation(n*Integral(a**2), (n, 0, 2)) == 3*Integral(a**2)
# test nested sum evaluation
s = Sum( Sum( Sum(2,(z,1,n+1)), (y,x+1,n)), (x,1,n))
assert 0 == (s.doit() - n*(n+1)*(n-1)).factor()
assert Sum(Sum(KroneckerDelta(m, n), (m, 1, 3)), (n, 1, 3)).doit() == 3
assert Sum(Sum(KroneckerDelta(k, m), (m, 1, 3)), (n, 1, 3)).doit() == \
3*Piecewise((1, And(S(1) <= k, k <= 3)), (0, True))
assert Sum(f(n)*Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, 3)).doit() == \
f(1) + f(2) + f(3)
assert Sum(f(n)*Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, oo)).doit() == \
Sum(Piecewise((f(n), n >= 0), (0, True)), (n, 1, oo))
l = Symbol('l', integer=True, positive=True)
assert Sum(f(l)*Sum(KroneckerDelta(m, l), (m, 0, oo)), (l, 1, oo)).doit() == \
Sum(f(l), (l, 1, oo))
# issue 2597
nmax = symbols('N', integer=True, positive=True)
pw = Piecewise((1, And(S(1) <= n, n <= nmax)), (0, True))
assert Sum(pw, (n, 1, nmax)).doit() == Sum(pw, (n, 1, nmax))
def test_separatevars():
x, y, z, n = symbols('x,y,z,n')
assert separatevars(2*n*x*z + 2*x*y*z) == 2*x*z*(n + y)
assert separatevars(x*z + x*y*z) == x*z*(1 + y)
assert separatevars(pi*x*z + pi*x*y*z) == pi*x*z*(1 + y)
assert separatevars(x*y**2*sin(x) + x*sin(x)*sin(y)) == \
x*(sin(y) + y**2)*sin(x)
assert separatevars(x*exp(x + y) + x*exp(x)) == x*(1 + exp(y))*exp(x)
assert separatevars((x*(y + 1))**z).is_Pow # != x**z*(1 + y)**z
assert separatevars(1 + x + y + x*y) == (x + 1)*(y + 1)
assert separatevars(y/pi*exp(-(z - x)/cos(n))) == \
y*exp(x/cos(n))*exp(-z/cos(n))/pi
assert separatevars((x + y)*(x - y) + y**2 + 2*x + 1) == (x + 1)**2
# issue 4858
p = Symbol('p', positive=True)
assert separatevars(sqrt(p**2 + x*p**2)) == p*sqrt(1 + x)
assert separatevars(sqrt(y*(p**2 + x*p**2))) == p*sqrt(y*(1 + x))
assert separatevars(sqrt(y*(p**2 + x*p**2)), force=True) == \
p*sqrt(y)*sqrt(1 + x)
# issue 4865
assert separatevars(sqrt(x*y)).is_Pow
assert separatevars(sqrt(x*y), force=True) == sqrt(x)*sqrt(y)
# issue 4957
# any type sequence for symbols is fine
assert separatevars(((2*x + 2)*y), dict=True, symbols=()) == \
{'coeff': 1, x: 2*x + 2, y: y}
# separable
assert separatevars(((2*x + 2)*y), dict=True, symbols=[x]) == \
{'coeff': y, x: 2*x + 2}
assert separatevars(((2*x + 2)*y), dict=True, symbols=[]) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True, symbols=None) == \
{'coeff': y*(2*x + 2)}
# not separable
assert separatevars(3, dict=True) is None
assert separatevars(2*x + y, dict=True, symbols=()) is None
assert separatevars(2*x + y, dict=True) is None
assert separatevars(2*x + y, dict=True, symbols=None) == {'coeff': 2*x + y}
# issue 4808
n, m = symbols('n,m', commutative=False)
assert separatevars(m + n*m) == (1 + n)*m
assert separatevars(x + x*n) == x*(1 + n)
# issue 4910
f = Function('f')
assert separatevars(f(x) + x*f(x)) == f(x) + x*f(x)
# a noncommutable object present
eq = x*(1 + hyper((), (), y*z))
assert separatevars(eq) == eq
def test_function_assumptions():
x = Symbol('x')
f = Function('f')
f_real = Function('f', real=True)
assert f != f_real
assert f(x) != f_real(x)
assert f(x).is_real is None
assert f_real(x).is_real is True
# Can also do it this way, but it won't be equal to f_real because of the
# way UndefinedFunction.__new__ works.
f_real2 = Function('f', is_real=True)
assert f_real2(x).is_real is True
def test_issue_6920():
e = [cos(x) + I*sin(x), cos(x) - I*sin(x),
cosh(x) - sinh(x), cosh(x) + sinh(x)]
ok = [exp(I*x), exp(-I*x), exp(-x), exp(x)]
# wrap in f to show that the change happens wherever ei occurs
f = Function('f')
assert [simplify(f(ei)).args[0] for ei in e] == ok
def test_klein_gordon_lagrangian():
m = Symbol("m")
phi = f(x, t)
L = -(diff(phi, t) ** 2 - diff(phi, x) ** 2 - m ** 2 * phi ** 2) / 2
eqna = Eq(diff(L, phi) - diff(L, diff(phi, x), x) - diff(L, diff(phi, t), t), 0)
eqnb = Eq(diff(phi, t, t) - diff(phi, x, x) + m ** 2 * phi, 0)
assert eqna == eqnb
def test_collect_D_0():
D = Derivative
f = Function('f')
x, a, b = symbols('x,a,b')
fxx = D(f(x), x, x)
# collect does not distinguish nested derivatives, so it returns
# -- (a + b)*D(D(f, x), x)
assert collect(a*fxx + b*fxx, fxx) == (a + b)*fxx
def test_collect_3():
"""Collect with respect to a product"""
a, b, c = symbols('a,b,c')
f = Function('f')
x, y, z, n = symbols('x,y,z,n')
assert collect(-x/8 + x*y, -x) == x*(y - S(1)/8)
assert collect( 1 + x*(y**2), x*y ) == 1 + x*(y**2)
assert collect( x*y + a*x*y, x*y) == x*y*(1 + a)
assert collect( 1 + x*y + a*x*y, x*y) == 1 + x*y*(1 + a)
assert collect(a*x*f(x) + b*(x*f(x)), x*f(x)) == x*(a + b)*f(x)
assert collect(a*x*log(x) + b*(x*log(x)), x*log(x)) == x*(a + b)*log(x)
assert collect(a*x**2*log(x)**2 + b*(x*log(x))**2, x*log(x)) == \
x**2*log(x)**2*(a + b)
# with respect to a product of three symbols
assert collect(y*x*z + a*x*y*z, x*y*z) == (1 + a)*x*y*z
def test_Sum_doit():
assert Sum(n * Integral(a ** 2), (n, 0, 2)).doit() == a ** 3
assert Sum(n * Integral(a ** 2), (n, 0, 2)).doit(deep=False) == 3 * Integral(a ** 2)
assert summation(n * Integral(a ** 2), (n, 0, 2)) == 3 * Integral(a ** 2)
# test nested sum evaluation
s = Sum(Sum(Sum(2, (z, 1, n + 1)), (y, x + 1, n)), (x, 1, n))
assert 0 == (s.doit() - n * (n + 1) * (n - 1)).factor()
assert Sum(KroneckerDelta(m, n), (m, -oo, oo)).doit() == Piecewise((1, And(-oo < n, n < oo)), (0, True))
assert Sum(x * KroneckerDelta(m, n), (m, -oo, oo)).doit() == Piecewise((x, And(-oo < n, n < oo)), (0, True))
assert Sum(Sum(KroneckerDelta(m, n), (m, 1, 3)), (n, 1, 3)).doit() == 3
assert Sum(Sum(KroneckerDelta(k, m), (m, 1, 3)), (n, 1, 3)).doit() == 3 * Piecewise(
(1, And(S(1) <= k, k <= 3)), (0, True)
)
assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, 3)).doit() == f(1) + f(2) + f(3)
assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, oo)).doit() == Sum(
Piecewise((f(n), And(Le(0, n), n < oo)), (0, True)), (n, 1, oo)
)
l = Symbol("l", integer=True, positive=True)
assert Sum(f(l) * Sum(KroneckerDelta(m, l), (m, 0, oo)), (l, 1, oo)).doit() == Sum(f(l), (l, 1, oo))
# issue 2597
nmax = symbols("N", integer=True, positive=True)
pw = Piecewise((1, And(S(1) <= n, n <= nmax)), (0, True))
assert Sum(pw, (n, 1, nmax)).doit() == Sum(pw, (n, 1, nmax))
q, s = symbols("q, s")
assert summation(1 / n ** (2 * s), (n, 1, oo)) == Piecewise(
(zeta(2 * s), 2 * s > 1), (Sum(n ** (-2 * s), (n, 1, oo)), True)
)
assert summation(1 / (n + 1) ** s, (n, 0, oo)) == Piecewise(
(zeta(s), s > 1), (Sum((n + 1) ** (-s), (n, 0, oo)), True)
)
assert summation(1 / (n + q) ** s, (n, 0, oo)) == Piecewise(
(zeta(s, q), And(q > 0, s > 1)), (Sum((n + q) ** (-s), (n, 0, oo)), True)
)
assert summation(1 / (n + q) ** s, (n, q, oo)) == Piecewise(
(zeta(s, 2 * q), And(2 * q > 0, s > 1)), (Sum((n + q) ** (-s), (n, q, oo)), True)
)
assert summation(1 / n ** 2, (n, 1, oo)) == zeta(2)
assert summation(1 / n ** s, (n, 0, oo)) == Sum(n ** (-s), (n, 0, oo))
def test_nargs():
f = Function('f')
assert f.nargs == S.Naturals0
assert f(1).nargs == S.Naturals0
assert Function('f', nargs=2)(1, 2).nargs == FiniteSet(2)
assert sin.nargs == FiniteSet(1)
assert sin(2).nargs == FiniteSet(1)
assert log.nargs == FiniteSet(1, 2)
assert log(2).nargs == FiniteSet(1, 2)
assert Function('f', nargs=2).nargs == FiniteSet(2)
assert Function('f', nargs=0).nargs == FiniteSet(0)
def test_sho_lagrangian():
m = Symbol('m')
k = Symbol('k')
x = f(t)
L = m*diff(x, t)**2/2 - k*x**2/2
eqna = Eq(diff(L, x), diff(L, diff(x, t), t))
eqnb = Eq(-k*x, m*diff(x, t, t))
assert eqna == eqnb
assert diff(L, x, t) == diff(L, t, x)
assert diff(L, diff(x, t), t) == m*diff(x, t, 2)
assert diff(L, t, diff(x, t)) == -k*x + m*diff(x, t, 2)
def test_issue_7231():
from sympy.abc import a
ans1 = f(x).series(x, a)
_xi_1 = ans1.atoms(Dummy).pop()
res = (f(a) + (-a + x)*Subs(Derivative(f(_xi_1), _xi_1), (_xi_1,), (a,)) +
(-a + x)**2*Subs(Derivative(f(_xi_1), _xi_1, _xi_1), (_xi_1,), (a,))/2 +
(-a + x)**3*Subs(Derivative(f(_xi_1), _xi_1, _xi_1, _xi_1),
(_xi_1,), (a,))/6 +
(-a + x)**4*Subs(Derivative(f(_xi_1), _xi_1, _xi_1, _xi_1, _xi_1),
(_xi_1,), (a,))/24 +
(-a + x)**5*Subs(Derivative(f(_xi_1), _xi_1, _xi_1, _xi_1, _xi_1, _xi_1),
(_xi_1,), (a,))/120 + O((-a + x)**6, (x, a)))
assert res == ans1
ans2 = f(x).series(x, a)
assert res == ans2
def test_issue_13843():
x = symbols('x')
f = Function('f')
m, n = symbols('m n', integer=True)
assert Derivative(Derivative(f(x), (x, m)), (x, n)) == Derivative(f(x), (x, m + n))
assert Derivative(Derivative(f(x), (x, m+5)), (x, n+3)) == Derivative(f(x), (x, m + n + 8))
assert Derivative(f(x), (x, n)).doit() == Derivative(f(x), (x, n))
def test_nargs():
f = Function('f')
assert f.nargs == S.Naturals0
assert f(1).nargs == S.Naturals0
assert Function('f', nargs=2)(1, 2).nargs == FiniteSet(2)
assert sin.nargs == FiniteSet(1)
assert sin(2).nargs == FiniteSet(1)
assert log.nargs == FiniteSet(1, 2)
assert log(2).nargs == FiniteSet(1, 2)
assert Function('f', nargs=2).nargs == FiniteSet(2)
assert Function('f', nargs=0).nargs == FiniteSet(0)
assert Function('f', nargs=(0, 1)).nargs == FiniteSet(0, 1)
assert Function('f', nargs=None).nargs == S.Naturals0
raises(ValueError, lambda: Function('f', nargs=()))
def test_function_subs():
f = Function("f")
S = Sum(x*f(y),(x,0,oo),(y,0,oo))
assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
assert S.subs(f(x),x) == S
raises(ValueError, lambda: S.subs(f(y),x+y) )
S = Sum(x*log(y),(x,0,oo),(y,0,oo))
assert S.subs(log(y),y) == S
S = Sum(x*f(y),(x,0,oo),(y,0,oo))
assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
def test_telescopic_sums():
# checks also input 2 of comment 1 issue 1028
assert Sum(1 / k - 1 / (k + 1), (k, 1, n)).doit() == 1 - 1 / (1 + n)
f = Function("f")
assert Sum(f(k) - f(k + 2), (k, m, n)).doit() == -f(1 + n) - f(2 + n) + f(m) + f(1 + m)
assert Sum(cos(k) - cos(k + 3), (k, 1, n)).doit() == -cos(1 + n) - cos(2 + n) - cos(3 + n) + cos(1) + cos(2) + cos(
3
)
# dummy variable shouldn't matter
assert telescopic(1 / m, -m / (1 + m), (m, n - 1, n)) == telescopic(1 / k, -k / (1 + k), (k, n - 1, n))
assert Sum(1 / x / (x - 1), (x, a, b)).doit() == -((a - b - 1) / (b * (a - 1)))
def test_simplify_function_inverse():
# "inverse" attribute does not guarantee that f(g(x)) is x
# so this simplification should not happen automatically.
# See issue #12140
x, y = symbols('x, y')
g = Function('g')
class f(Function):
def inverse(self, argindex=1):
return g
assert simplify(f(g(x))) == f(g(x))
assert inversecombine(f(g(x))) == x
assert simplify(f(g(x)), inverse=True) == x
assert simplify(f(g(sin(x)**2 + cos(x)**2)), inverse=True) == 1
assert simplify(f(g(x, y)), inverse=True) == f(g(x, y))
assert simplify(2*asin(sin(3*x)), inverse=True) == 6*x
def test_deriv_wrt_function():
x = f(t)
xd = diff(x, t)
xdd = diff(xd, t)
y = g(t)
yd = diff(y, t)
assert diff(x, t) == xd
assert diff(2 * x + 4, t) == 2 * xd
assert diff(2 * x + 4 + y, t) == 2 * xd + yd
assert diff(2 * x + 4 + y * x, t) == 2 * xd + x * yd + xd * y
assert diff(2 * x + 4 + y * x, x) == 2 + y
assert diff(4 * x ** 2 + 3 * x + x * y, t) == 3 * xd + x * yd + xd * y + 8 * x * xd
assert diff(4 * x ** 2 + 3 * xd + x * y, t) == 3 * xdd + x * yd + xd * y + 8 * x * xd
assert diff(4 * x ** 2 + 3 * xd + x * y, xd) == 3
assert diff(4 * x ** 2 + 3 * xd + x * y, xdd) == 0
assert diff(sin(x), t) == xd * cos(x)
assert diff(exp(x), t) == xd * exp(x)
assert diff(sqrt(x), t) == xd / (2 * sqrt(x))
def test_distribution_over_equality():
assert Product(Eq(x * 2, f(x)),
(x, 1, 3)).doit() == Eq(48,
f(1) * f(2) * f(3))
assert Sum(Eq(f(x), x**2), (x, 0, y)) == \
Eq(Sum(f(x), (x, 0, y)), Sum(x**2, (x, 0, y)))
def test_sort_variable():
vsort = Derivative._sort_variables
assert vsort((x, y, z)) == [x, y, z]
assert vsort((h(x), g(x), f(x))) == [f(x), g(x), h(x)]
assert vsort((z, y, x, h(x), g(x), f(x))) == [x, y, z, f(x), g(x), h(x)]
assert vsort((x, f(x), y, f(y))) == [x, f(x), y, f(y)]
assert vsort((y, x, g(x), f(x), z, h(x), y, x)) == \
[x, y, f(x), g(x), z, h(x), x, y]
assert vsort((z, y, f(x), x, f(x), g(x))) == [y, z, f(x), x, f(x), g(x)]
assert vsort((z, y, f(x), x, f(x), g(x), z, z, y, x)) == \
[y, z, f(x), x, f(x), g(x), x, y, z, z]
def test_collect_issues():
D = Derivative
f = Function('f')
e = (1 + x * D(f(x), x) + D(f(x), x)) / f(x)
assert collect(e.expand(), f(x).diff(x)) != e
def test_issue_13143():
fx = f(x).diff(x)
e = f(x) + fx + f(x) * fx
# collect function before derivative
assert collect(e, Wild('w')) == f(x) * (fx + 1) + fx
e = f(x) + f(x) * fx + x * fx * f(x)
assert collect(e, fx) == (x * f(x) + f(x)) * fx + f(x)
assert collect(e, f(x)) == (x * fx + fx + 1) * f(x)
e = f(x) + fx + f(x) * fx
assert collect(e, [f(x), fx]) == f(x) * (1 + fx) + fx
assert collect(e, [fx, f(x)]) == fx * (1 + f(x)) + f(x)
def test_derivative_evaluate():
assert Derivative(sin(x), x) != diff(sin(x), x)
assert Derivative(sin(x), x).doit() == diff(sin(x), x)
assert Derivative(Derivative(f(x), x), x) == diff(f(x), x, x)
assert Derivative(sin(x), x, 0) == sin(x)
def test_derivative_subs_self_bug():
d = diff(f(x), x)
assert d.subs(d, y) == y
def test_karr_convention():
# Test the Karr summation convention that we want to hold.
# See his paper "Summation in Finite Terms" for a detailed
# reasoning why we really want exactly this definition.
# The convention is described on page 309 and essentially
# in section 1.4, definition 3:
#
# \sum_{m <= i < n} f(i) 'has the obvious meaning' for m < n
# \sum_{m <= i < n} f(i) = 0 for m = n
# \sum_{m <= i < n} f(i) = - \sum_{n <= i < m} f(i) for m > n
#
# It is important to note that he defines all sums with
# the upper limit being *exclusive*.
# In contrast, sympy and the usual mathematical notation has:
#
# sum_{i = a}^b f(i) = f(a) + f(a+1) + ... + f(b-1) + f(b)
#
# with the upper limit *inclusive*. So translating between
# the two we find that:
#
# \sum_{m <= i < n} f(i) = \sum_{i = m}^{n-1} f(i)
#
# where we intentionally used two different ways to typeset the
# sum and its limits.
i = Symbol("i", integer=True)
k = Symbol("k", integer=True)
j = Symbol("j", integer=True)
# A simple example with a concrete summand and symbolic limits.
# The normal sum: m = k and n = k + j and therefore m < n:
m = k
n = k + j
a = m
b = n - 1
S1 = Sum(i**2, (i, a, b)).doit()
# The reversed sum: m = k + j and n = k and therefore m > n:
m = k + j
n = k
a = m
b = n - 1
S2 = Sum(i**2, (i, a, b)).doit()
assert simplify(S1 + S2) == 0
# Test the empty sum: m = k and n = k and therefore m = n:
m = k
n = k
a = m
b = n - 1
Sz = Sum(i**2, (i, a, b)).doit()
assert Sz == 0
# Another example this time with an unspecified summand and
# numeric limits. (We can not do both tests in the same example.)
f = Function("f")
# The normal sum with m < n:
m = 2
n = 11
a = m
b = n - 1
S1 = Sum(f(i), (i, a, b)).doit()
# The reversed sum with m > n:
m = 11
n = 2
a = m
b = n - 1
S2 = Sum(f(i), (i, a, b)).doit()
assert simplify(S1 + S2) == 0
# Test the empty sum with m = n:
m = 5
n = 5
a = m
b = n - 1
Sz = Sum(f(i), (i, a, b)).doit()
assert Sz == 0
def test_diff_wrt_func_subs():
assert f(g(x)).diff(x).subs(g, Lambda(x, 2 * x)).doit() == f(2 * x).diff(x)
def test_simplify_expr():
x, y, z, k, n, m, w, s, A = symbols('x,y,z,k,n,m,w,s,A')
f = Function('f')
assert all(simplify(tmp) == tmp for tmp in [I, E, oo, x, -x, -oo, -E, -I])
e = 1 / x + 1 / y
assert e != (x + y) / (x * y)
assert simplify(e) == (x + y) / (x * y)
e = A**2 * s**4 / (4 * pi * k * m**3)
assert simplify(e) == e
e = (4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)
assert simplify(e) == 0
e = (-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y) / (x + y)**2
assert simplify(e) == -2 * y
e = -x - y - (x + y)**(-1) * y**2 + (x + y)**(-1) * x**2
assert simplify(e) == -2 * y
e = (x + x * y) / x
assert simplify(e) == 1 + y
e = (f(x) + y * f(x)) / f(x)
assert simplify(e) == 1 + y
e = (2 * (1 / n - cos(n * pi) / n)) / pi
assert simplify(e) == (-cos(pi * n) + 1) / (pi * n) * 2
e = integrate(1 / (x**3 + 1), x).diff(x)
assert simplify(e) == 1 / (x**3 + 1)
e = integrate(x / (x**2 + 3 * x + 1), x).diff(x)
assert simplify(e) == x / (x**2 + 3 * x + 1)
f = Symbol('f')
A = Matrix([[2 * k - m * w**2, -k], [-k, k - m * w**2]]).inv()
assert simplify((A * Matrix([0, f]))[1] - (-f * (2 * k - m * w**2) /
(k**2 - (k - m * w**2) *
(2 * k - m * w**2)))) == 0
f = -x + y / (z + t) + z * x / (z + t) + z * a / (z + t) + t * x / (z + t)
assert simplify(f) == (y + a * z) / (z + t)
# issue 10347
expr = -x * (y**2 - 1) * (
2 * y**2 * (x**2 - 1) / (a * (x**2 - y**2)**2) + (x**2 - 1) /
(a * (x**2 - y**2))) / (a * (x**2 - y**2)) + x * (
-2 * x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - y**2)**2) -
x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - 1) * (x**2 - y**2)) +
(x**2 * sqrt((-x**2 + 1) * (y**2 - 1)) *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) / (x**2 - 1) + sqrt(
(-x**2 + 1) * (y**2 - 1)) *
(x * (-x * y**2 + x) / sqrt(-x**2 * y**2 + x**2 + y**2 - 1) +
sqrt(-x**2 * y**2 + x**2 + y**2 - 1)) * sin(z)) / (a * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (x**2 - y**2))
) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) / (
a * (x**2 - y**2)) + x * (
-2 * x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(a * (x**2 - y**2)**2) -
x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(a * (x**2 - 1) * (x**2 - y**2)) +
(x**2 * sqrt((-x**2 + 1) *
(y**2 - 1)) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1)
* cos(z) / (x**2 - 1) + x * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (-x * y**2 + x) * cos(z) /
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) + sqrt(
(-x**2 + 1) *
(y**2 - 1)) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) *
cos(z)) / (a * sqrt((-x**2 + 1) * (y**2 - 1)) * (x**2 - y**2))
) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) / (
a *
(x**2 - y**2)) - y * sqrt((-x**2 + 1) * (y**2 - 1)) * (
-x * y * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - y**2) *
(y**2 - 1)) +
2 * x * y * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - y**2)**2) +
(x * y * sqrt((-x**2 + 1) * (y**2 - 1)) *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(y**2 - 1) + x * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (-x**2 * y + y) * sin(z) /
sqrt(-x**2 * y**2 + x**2 + y**2 - 1)) / (a * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (x**2 - y**2))
) * sin(z) / (a * (x**2 - y**2)) + y * (x**2 - 1) * (
-2 * x * y *
(x**2 - 1) /
(a * (x**2 - y**2)**2) + 2 * x * y / (a * (x**2 - y**2))
) / (a *
(x**2 - y**2)) + y * (x**2 - 1) * (y**2 - 1) * (
-x * y * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) *
cos(z) / (a * (x**2 - y**2) *
(y**2 - 1)) + 2 * x * y *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(a *
(x**2 - y**2)**2) +
(x * y * sqrt((-x**2 + 1) * (y**2 - 1)) *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(y**2 - 1) + x * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (-x**2 * y + y) *
cos(z) / sqrt(-x**2 * y**2 + x**2 + y**2 - 1)) /
(a * sqrt((-x**2 + 1) * (y**2 - 1)) *
(x**2 - y**2))) * cos(z) / (a * sqrt(
(-x**2 + 1) *
(y**2 - 1)) * (x**2 - y**2)) - x * sqrt(
(-x**2 + 1) * (y**2 - 1)
) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(
z)**2 / (a**2 * (x**2 - 1) * (x**2 - y**2) *
(y**2 - 1)) - x * sqrt(
(-x**2 + 1) *
(y**2 - 1)) * sqrt(
-x**2 * y**2 + x**2 + y**2 -
1) * cos(z)**2 / (
a**2 * (x**2 - 1) *
(x**2 - y**2) *
(y**2 - 1))
assert simplify(expr) == 2 * x / (a**2 * (x**2 - y**2))
#issue 17631
assert simplify('((-1/2)*Boole(True)*Boole(False)-1)*Boole(True)') == \
Mul(sympify('(2 + Boole(True)*Boole(False))'), sympify('-Boole(True)/2'))
A, B = symbols('A,B', commutative=False)
assert simplify(A * B - B * A) == A * B - B * A
assert simplify(A / (1 + y / x)) == x * A / (x + y)
assert simplify(A * (1 / x + 1 / y)) == A / x + A / y #(x + y)*A/(x*y)
assert simplify(log(2) + log(3)) == log(6)
assert simplify(log(2 * x) - log(2)) == log(x)
assert simplify(hyper([], [], x)) == exp(x)
def test_collect_D():
D = Derivative
f = Function('f')
x, a, b = symbols('x,a,b')
fx = D(f(x), x)
fxx = D(f(x), x, x)
assert collect(a * fx + b * fx, fx) == (a + b) * fx
assert collect(a * D(fx, x) + b * D(fx, x), fx) == (a + b) * D(fx, x)
assert collect(a * fxx + b * fxx, fx) == (a + b) * D(fx, x)
# issue 4784
assert collect(5 * f(x) + 3 * fx, fx) == 5 * f(x) + 3 * fx
assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x)) == \
(x*f(x) + f(x))*D(f(x), x) + f(x)
assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x), exact=True) == \
(x*f(x) + f(x))*D(f(x), x) + f(x)
assert collect(1/f(x) + 1/f(x)*diff(f(x), x) + x*diff(f(x), x)/f(x), f(x).diff(x), exact=True) == \
(1/f(x) + x/f(x))*D(f(x), x) + 1/f(x)
def test_issue_4711():
x = Symbol("x")
assert Symbol('f')(x) == f(x)
def test_evalf_symbolic():
f, g = symbols('f g', cls=Function)
# issue 6328
expr = Sum(f(x), (x, 1, 3)) + Sum(g(x), (x, 1, 3))
assert expr.evalf() == expr
def test_derivative_linearity():
assert diff(-f(x), x) == -diff(f(x), x)
assert diff(8 * f(x), x) == 8 * diff(f(x), x)
assert diff(8 * f(x), x) != 7 * diff(f(x), x)
assert diff(8 * f(x) * x, x) == 8 * f(x) + 8 * x * diff(f(x), x)
assert diff(8 * f(x) * y * x,
x) == 8 * y * f(x) + 8 * y * x * diff(f(x), x)
def test_powsimp():
x, y, z, n = symbols('x,y,z,n')
f = Function('f')
assert powsimp(4**x * 2**(-x) * 2**(-x)) == 1
assert powsimp((-4)**x * (-2)**(-x) * 2**(-x)) == 1
assert powsimp(f(4**x * 2**(-x) * 2**(-x))) == f(4**x * 2**(-x) * 2**(-x))
assert powsimp(f(4**x * 2**(-x) * 2**(-x)), deep=True) == f(1)
assert exp(x) * exp(y) == exp(x) * exp(y)
assert powsimp(exp(x) * exp(y)) == exp(x + y)
assert powsimp(exp(x) * exp(y) * 2**x * 2**y) == (2 * E)**(x + y)
assert powsimp(exp(x)*exp(y)*2**x*2**y, combine='exp') == \
exp(x + y)*2**(x + y)
assert powsimp(exp(x)*exp(y)*exp(2)*sin(x) + sin(y) + 2**x*2**y) == \
exp(2 + x + y)*sin(x) + sin(y) + 2**(x + y)
assert powsimp(sin(exp(x) * exp(y))) == sin(exp(x) * exp(y))
assert powsimp(sin(exp(x) * exp(y)), deep=True) == sin(exp(x + y))
assert powsimp(x**2 * x**y) == x**(2 + y)
# This should remain factored, because 'exp' with deep=True is supposed
# to act like old automatic exponent combining.
assert powsimp((1 + E*exp(E))*exp(-E), combine='exp', deep=True) == \
(1 + exp(1 + E))*exp(-E)
assert powsimp((1 + E*exp(E))*exp(-E), deep=True) == \
(1 + exp(1 + E))*exp(-E)
assert powsimp((1 + E * exp(E)) * exp(-E)) == (1 + exp(1 + E)) * exp(-E)
assert powsimp((1 + E*exp(E))*exp(-E), combine='exp') == \
(1 + exp(1 + E))*exp(-E)
assert powsimp((1 + E*exp(E))*exp(-E), combine='base') == \
(1 + E*exp(E))*exp(-E)
x, y = symbols('x,y', nonnegative=True)
n = Symbol('n', real=True)
assert powsimp(y**n * (y / x)**(-n)) == x**n
assert powsimp(x**(x**(x*y)*y**(x*y))*y**(x**(x*y)*y**(x*y)), deep=True) \
== (x*y)**(x*y)**(x*y)
assert powsimp(2**(2**(2 * x) * x), deep=False) == 2**(2**(2 * x) * x)
assert powsimp(2**(2**(2 * x) * x), deep=True) == 2**(x * 4**x)
assert powsimp(
exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \
exp(-x + exp(-x)*exp(-x*log(x)))
assert powsimp(
exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \
exp(-x + exp(-x)*exp(-x*log(x)))
assert powsimp((x + y) / (3 * z), deep=False,
combine='exp') == (x + y) / (3 * z)
assert powsimp((x / 3 + y / 3) / z, deep=True,
combine='exp') == (x / 3 + y / 3) / z
assert powsimp(exp(x)/(1 + exp(x)*exp(y)), deep=True) == \
exp(x)/(1 + exp(x + y))
assert powsimp(x * y**(z**x * z**y), deep=True) == x * y**(z**(x + y))
assert powsimp((z**x * z**y)**x, deep=True) == (z**(x + y))**x
assert powsimp(x * (z**x * z**y)**x, deep=True) == x * (z**(x + y))**x
p = symbols('p', positive=True)
assert powsimp((1 / x)**log(2) / x) == (1 / x)**(1 + log(2))
assert powsimp((1 / p)**log(2) / p) == p**(-1 - log(2))
# coefficient of exponent can only be simplified for positive bases
assert powsimp(2**(2 * x)) == 4**x
assert powsimp((-1)**(2 * x)) == (-1)**(2 * x)
i = symbols('i', integer=True)
assert powsimp((-1)**(2 * i)) == 1
assert powsimp((-1)**(-x)) != (-1)**x # could be 1/((-1)**x), but is not
# force=True overrides assumptions
assert powsimp((-1)**(2 * x), force=True) == 1
# rational exponents allow combining of negative terms
w, n, m = symbols('w n m', negative=True)
e = i / a # not a rational exponent if `a` is unknown
ex = w**e * n**e * m**e
assert powsimp(ex) == m**(i / a) * n**(i / a) * w**(i / a)
e = i / 3
ex = w**e * n**e * m**e
assert powsimp(ex) == (-1)**i * (-m * n * w)**(i / 3)
e = (3 + i) / i
ex = w**e * n**e * m**e
assert powsimp(ex) == (-1)**(3 * e) * (-m * n * w)**e
eq = x**(2 * a / 3)
# eq != (x**a)**(2/3) (try x = -1 and a = 3 to see)
assert powsimp(eq).exp == eq.exp == 2 * a / 3
# powdenest goes the other direction
assert powsimp(2**(2 * x)) == 4**x
assert powsimp(exp(p / 2)) == exp(p / 2)
# issue 6368
eq = Mul(*[sqrt(Dummy(imaginary=True)) for i in range(3)])
assert powsimp(eq) == eq and eq.is_Mul
assert all(powsimp(e) == e for e in (sqrt(x**a), sqrt(x**2)))
# issue 8836
assert str(powsimp(exp(I * pi / 3) * root(-1, 3))) == '(-1)**(2/3)'
# issue 9183
assert powsimp(-0.1**x) == -0.1**x
# issue 10095
assert powsimp((1 / (2 * E))**oo) == (exp(-1) / 2)**oo
# PR 13131
eq = sin(2 * x)**2 * sin(2.0 * x)**2
assert powsimp(eq) == eq
# issue 14615
assert powsimp(x**2 * y**3 *
(x * y**2)**(S(3) / 2)) == x * y * (x * y**2)**(S(5) / 2)
def test_separatevars():
x, y, z, n = symbols('x,y,z,n')
assert separatevars(2*n*x*z + 2*x*y*z) == 2*x*z*(n + y)
assert separatevars(x*z + x*y*z) == x*z*(1 + y)
assert separatevars(pi*x*z + pi*x*y*z) == pi*x*z*(1 + y)
assert separatevars(x*y**2*sin(x) + x*sin(x)*sin(y)) == \
x*(sin(y) + y**2)*sin(x)
assert separatevars(x*exp(x + y) + x*exp(x)) == x*(1 + exp(y))*exp(x)
assert separatevars((x*(y + 1))**z).is_Pow # != x**z*(1 + y)**z
assert separatevars(1 + x + y + x*y) == (x + 1)*(y + 1)
assert separatevars(y/pi*exp(-(z - x)/cos(n))) == \
y*exp(x/cos(n))*exp(-z/cos(n))/pi
assert separatevars((x + y)*(x - y) + y**2 + 2*x + 1) == (x + 1)**2
# issue 4858
p = Symbol('p', positive=True)
assert separatevars(sqrt(p**2 + x*p**2)) == p*sqrt(1 + x)
assert separatevars(sqrt(y*(p**2 + x*p**2))) == p*sqrt(y*(1 + x))
assert separatevars(sqrt(y*(p**2 + x*p**2)), force=True) == \
p*sqrt(y)*sqrt(1 + x)
# issue 4865
assert separatevars(sqrt(x*y)).is_Pow
assert separatevars(sqrt(x*y), force=True) == sqrt(x)*sqrt(y)
# issue 4957
# any type sequence for symbols is fine
assert separatevars(((2*x + 2)*y), dict=True, symbols=()) == \
{'coeff': 1, x: 2*x + 2, y: y}
# separable
assert separatevars(((2*x + 2)*y), dict=True, symbols=[x]) == \
{'coeff': y, x: 2*x + 2}
assert separatevars(((2*x + 2)*y), dict=True, symbols=[]) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True, symbols=None) == \
{'coeff': y*(2*x + 2)}
# not separable
assert separatevars(3, dict=True) is None
assert separatevars(2*x + y, dict=True, symbols=()) is None
assert separatevars(2*x + y, dict=True) is None
assert separatevars(2*x + y, dict=True, symbols=None) == {'coeff': 2*x + y}
# issue 4808
n, m = symbols('n,m', commutative=False)
assert separatevars(m + n*m) == (1 + n)*m
assert separatevars(x + x*n) == x*(1 + n)
# issue 4910
f = Function('f')
assert separatevars(f(x) + x*f(x)) == f(x) + x*f(x)
# a noncommutable object present
eq = x*(1 + hyper((), (), y*z))
assert separatevars(eq) == eq
s = separatevars(abs(x*y))
assert s == abs(x)*abs(y) and s.is_Mul
z = cos(1)**2 + sin(1)**2 - 1
a = abs(x*z)
s = separatevars(a)
assert not a.is_Mul and s.is_Mul and s == abs(x)*abs(z)
s = separatevars(abs(x*y*z))
assert s == abs(x)*abs(y)*abs(z)
# abs(x+y)/abs(z) would be better but we test this here to
# see that it doesn't raise
assert separatevars(abs((x+y)/z)) == abs((x+y)/z)
def test_diff_wrt():
fx = f(x)
dfx = diff(f(x), x)
ddfx = diff(f(x), x, x)
assert diff(sin(fx) + fx**2, fx) == cos(fx) + 2 * fx
assert diff(sin(dfx) + dfx**2, dfx) == cos(dfx) + 2 * dfx
assert diff(sin(ddfx) + ddfx**2, ddfx) == cos(ddfx) + 2 * ddfx
assert diff(fx**2, dfx) == 0
assert diff(fx**2, ddfx) == 0
assert diff(dfx**2, fx) == 0
assert diff(dfx**2, ddfx) == 0
assert diff(ddfx**2, dfx) == 0
assert diff(fx * dfx * ddfx, fx) == dfx * ddfx
assert diff(fx * dfx * ddfx, dfx) == fx * ddfx
assert diff(fx * dfx * ddfx, ddfx) == fx * dfx
assert diff(f(x), x).diff(f(x)) == 0
assert (sin(f(x)) - cos(diff(f(x), x))).diff(f(x)) == cos(f(x))
assert diff(sin(fx), fx, x) == diff(sin(fx), x, fx)
# Chain rule cases
assert f(g(x)).diff(x) == \
Subs(Derivative(f(x), x), (x,), (g(x),))*Derivative(g(x), x)
assert diff(f(g(x), h(x)), x) == \
Subs(Derivative(f(y, h(x)), y), (y,), (g(x),))*Derivative(g(x), x) + \
Subs(Derivative(f(g(x), y), y), (y,), (h(x),))*Derivative(h(x), x)
assert f(sin(x)).diff(x) == Subs(Derivative(f(x), x), (x, ),
(sin(x), )) * cos(x)
assert diff(f(g(x)), g(x)) == Subs(Derivative(f(x), x), (x, ), (g(x), ))
def test_Derivative_as_finite_difference():
# Central 1st derivative at gridpoint
x, h = symbols('x h', real=True)
dfdx = f(x).diff(x)
assert (dfdx.as_finite_difference([x - 2, x - 1, x, x + 1, x + 2]) -
(S(1) / 12 * (f(x - 2) - f(x + 2)) + S(2) / 3 *
(f(x + 1) - f(x - 1)))).simplify() == 0
# Central 1st derivative "half-way"
assert (dfdx.as_finite_difference() -
(f(x + S(1) / 2) - f(x - S(1) / 2))).simplify() == 0
assert (dfdx.as_finite_difference(h) -
(f(x + h / S(2)) - f(x - h / S(2))) / h).simplify() == 0
assert (dfdx.as_finite_difference([x - 3 * h, x - h, x + h, x + 3 * h]) -
(S(9) / (8 * 2 * h) * (f(x + h) - f(x - h)) + S(1) / (24 * 2 * h) *
(f(x - 3 * h) - f(x + 3 * h)))).simplify() == 0
# One sided 1st derivative at gridpoint
assert (dfdx.as_finite_difference([0, 1, 2], 0) -
(-S(3) / 2 * f(0) + 2 * f(1) - f(2) / 2)).simplify() == 0
assert (dfdx.as_finite_difference([x, x + h], x) -
(f(x + h) - f(x)) / h).simplify() == 0
assert (dfdx.as_finite_difference([x - h, x, x + h], x - h) -
(-S(3) / (2 * h) * f(x - h) + 2 / h * f(x) - S(1) /
(2 * h) * f(x + h))).simplify() == 0
# One sided 1st derivative "half-way"
assert (
dfdx.as_finite_difference(
[x - h, x + h, x + 3 * h, x + 5 * h, x + 7 * h]) - 1 / (2 * h) *
(-S(11) / (12) * f(x - h) + S(17) /
(24) * f(x + h) + S(3) / 8 * f(x + 3 * h) - S(5) / 24 * f(x + 5 * h) +
S(1) / 24 * f(x + 7 * h))).simplify() == 0
d2fdx2 = f(x).diff(x, 2)
# Central 2nd derivative at gridpoint
assert (d2fdx2.as_finite_difference([x - h, x, x + h]) - h**-2 *
(f(x - h) + f(x + h) - 2 * f(x))).simplify() == 0
assert (
d2fdx2.as_finite_difference([x - 2 * h, x - h, x, x + h, x + 2 * h]) -
h**-2 * (-S(1) / 12 * (f(x - 2 * h) + f(x + 2 * h)) + S(4) / 3 *
(f(x + h) + f(x - h)) - S(5) / 2 * f(x))).simplify() == 0
# Central 2nd derivative "half-way"
assert (d2fdx2.as_finite_difference([x - 3 * h, x - h, x + h, x + 3 * h]) -
(2 * h)**-2 * (S(1) / 2 *
(f(x - 3 * h) + f(x + 3 * h)) - S(1) / 2 *
(f(x + h) + f(x - h)))).simplify() == 0
# One sided 2nd derivative at gridpoint
assert (d2fdx2.as_finite_difference([x, x + h, x + 2 * h, x + 3 * h]) -
h**-2 * (2 * f(x) - 5 * f(x + h) + 4 * f(x + 2 * h) - f(x + 3 * h))
).simplify() == 0
# One sided 2nd derivative at "half-way"
assert (
d2fdx2.as_finite_difference([x - h, x + h, x + 3 * h, x + 5 * h]) -
(2 * h)**-2 *
(S(3) / 2 * f(x - h) - S(7) / 2 * f(x + h) + S(5) / 2 * f(x + 3 * h) -
S(1) / 2 * f(x + 5 * h))).simplify() == 0
d3fdx3 = f(x).diff(x, 3)
# Central 3rd derivative at gridpoint
assert (d3fdx3.as_finite_difference() -
(-f(x - 3 / S(2)) + 3 * f(x - 1 / S(2)) - 3 * f(x + 1 / S(2)) +
f(x + 3 / S(2)))).simplify() == 0
assert (d3fdx3.as_finite_difference(
[x - 3 * h, x - 2 * h, x - h, x, x + h, x + 2 * h, x + 3 * h]) -
h**-3 * (S(1) / 8 * (f(x - 3 * h) - f(x + 3 * h)) - f(x - 2 * h) +
f(x + 2 * h) + S(13) / 8 *
(f(x - h) - f(x + h)))).simplify() == 0
# Central 3rd derivative at "half-way"
assert (d3fdx3.as_finite_difference([x - 3 * h, x - h, x + h, x + 3 * h]) -
(2 * h)**-3 * (f(x + 3 * h) - f(x - 3 * h) + 3 *
(f(x - h) - f(x + h)))).simplify() == 0
# One sided 3rd derivative at gridpoint
assert (d3fdx3.as_finite_difference([x, x + h, x + 2 * h, x + 3 * h]) -
h**-3 * (f(x + 3 * h) - f(x) + 3 *
(f(x + h) - f(x + 2 * h)))).simplify() == 0
# One sided 3rd derivative at "half-way"
assert (d3fdx3.as_finite_difference([x - h, x + h, x + 3 * h, x + 5 * h]) -
(2 * h)**-3 * (f(x + 5 * h) - f(x - h) + 3 *
(f(x + h) - f(x + 3 * h)))).simplify() == 0
# issue 11007
y = Symbol('y', real=True)
d2fdxdy = f(x, y).diff(x, y)
ref0 = Derivative(f(x + S(1) / 2, y), y) - Derivative(
f(x - S(1) / 2, y), y)
assert (d2fdxdy.as_finite_difference(wrt=x) - ref0).simplify() == 0
half = S(1) / 2
xm, xp, ym, yp = x - half, x + half, y - half, y + half
ref2 = f(xm, ym) + f(xp, yp) - f(xp, ym) - f(xm, yp)
assert (d2fdxdy.as_finite_difference() - ref2).simplify() == 0
def test_issue_4194():
# simplify should call cancel
from sympy.abc import x, y
f = Function('f')
assert simplify((4 * x + 6 * f(y)) / (2 * x + 3 * f(y))) == 2
def test_diff_wrt_value():
assert Expr()._diff_wrt is False
assert x._diff_wrt is True
assert f(x)._diff_wrt is True
assert Derivative(f(x), x)._diff_wrt is True
assert Derivative(x**2, x)._diff_wrt is False
("\\frac{a}{b}", a / b),
("\\frac{a + b}{c}", _Mul(a + b, _Pow(c, -1))),
("\\frac{7}{3}", _Mul(7, _Pow(3, -1))),
("(\\csc x)(\\sec y)", csc(x) * sec(y)),
("\\lim_{x \\to 3} a", Limit(a, x, 3)),
("\\lim_{x \\rightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\Rightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\longrightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\Longrightarrow 3} a", Limit(a, x, 3)),
("\\lim_{x \\to 3^{+}} a", Limit(a, x, 3, dir='+')),
("\\lim_{x \\to 3^{-}} a", Limit(a, x, 3, dir='-')),
("\\infty", oo),
("\\lim_{x \\to \\infty} \\frac{1}{x}",
Limit(_Mul(1, _Pow(x, -1)), x, oo)),
("\\frac{d}{dx} x", Derivative(x, x)),
("\\frac{d}{dt} x", Derivative(x, t)), ("f(x)", f(x)),
("f(x, y)", f(x, y)), ("f(x, y, z)", f(x, y, z)),
("\\frac{d f(x)}{dx}", Derivative(f(x), x)),
("\\frac{d\\theta(x)}{dx}", Derivative(theta(x), x)),
("|x|", _Abs(x)), ("||x||", _Abs(Abs(x))),
("|x||y|", _Abs(x) * _Abs(y)),
("||x||y||", _Abs(_Abs(x) * _Abs(y))),
("\\pi^{|xy|}", Symbol('pi')**_Abs(x * y)),
("\\int x dx", Integral(x, x)),
("\\int x d\\theta", Integral(x, theta)),
("\\int (x^2 - y)dx", Integral(x**2 - y, x)),
("\\int x + a dx", Integral(_Add(x, a), x)),
("\\int da", Integral(1, a)),
("\\int_0^7 dx", Integral(1, (x, 0, 7))),
("\\int_a^b x dx", Integral(x, (x, a, b))),
("\\int^b_a x dx", Integral(x, (x, a, b))),
def test_deriv1():
# These all requre derivatives evaluated at a point (issue 4719) to work.
# See issue 4624
assert f(
2 * x).diff(x) == 2 * Subs(Derivative(f(x), x), Tuple(x), Tuple(2 * x))
assert (f(x)**3).diff(x) == 3 * f(x)**2 * f(x).diff(x)
assert (f(2 * x)**3).diff(x) == 6 * f(2 * x)**2 * Subs(
Derivative(f(x), x), Tuple(x), Tuple(2 * x))
assert f(2 + x).diff(x) == Subs(Derivative(f(x), x), Tuple(x),
Tuple(x + 2))
assert f(2 + 3 * x).diff(x) == 3 * Subs(Derivative(f(x), x), Tuple(x),
Tuple(3 * x + 2))
assert f(3 * sin(x)).diff(x) == 3 * cos(x) * Subs(Derivative(
f(x), x), Tuple(x), Tuple(3 * sin(x)))
# See issue 8510
assert f(x, x + z).diff(x) == Subs(Derivative(f(y, x + z), y), Tuple(y), Tuple(x)) \
+ Subs(Derivative(f(x, y), y), Tuple(y), Tuple(x + z))
assert f(x, x**2).diff(x) == Subs(Derivative(f(y, x**2), y), Tuple(y), Tuple(x)) \
+ 2*x*Subs(Derivative(f(x, y), y), Tuple(y), Tuple(x**2))
def test_issue_10555():
f = Function('f')
assert solveset(f(x) - pi/2, x, S.Reals) == \
ConditionSet(x, Eq(2*f(x) - pi, 0), S.Reals)
def test_Subs2():
# this reflects a limitation of subs(), probably won't fix
assert Subs(f(x), x**2, x).doit() == f(sqrt(x))
def test_collect_Wild():
"""Collect with respect to functions with Wild argument"""
a, b, x, y = symbols('a b x y')
f = Function('f')
w1 = Wild('.1')
w2 = Wild('.2')
assert collect(f(x) + a * f(x), f(w1)) == (1 + a) * f(x)
assert collect(f(x, y) + a * f(x, y), f(w1)) == f(x, y) + a * f(x, y)
assert collect(f(x, y) + a * f(x, y), f(w1, w2)) == (1 + a) * f(x, y)
assert collect(f(x, y) + a * f(x, y), f(w1, w1)) == f(x, y) + a * f(x, y)
assert collect(f(x, x) + a * f(x, x), f(w1, w1)) == (1 + a) * f(x, x)
assert collect(a * (x + 1)**y + (x + 1)**y, w1**y) == (1 + a) * (x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, w1**b) == \
a*(x + 1)**y + (x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, (x + 1)**w2) == \
(1 + a)*(x + 1)**y
assert collect(a * (x + 1)**y + (x + 1)**y, w1**w2) == (1 + a) * (x + 1)**y
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_diff_symbols():
assert diff(f(x, y, z), x, y, z) == Derivative(f(x, y, z), x, y, z)
assert diff(f(x, y, z), x, x, x) == Derivative(f(x, y, z), x, x, x)
assert diff(f(x, y, z), x, 3) == Derivative(f(x, y, z), x, 3)
# issue 5028
assert [diff(-z + x / y, sym)
for sym in (z, x, y)] == [-1, 1 / y, -x / y**2]
assert diff(f(x, y, z), x, y, z, 2) == Derivative(f(x, y, z), x, y, z, z)
assert diff(f(x, y, z), x, y, z, 2, evaluate=False) == \
Derivative(f(x, y, z), x, y, z, z)
assert Derivative(f(x, y, z), x, y, z)._eval_derivative(z) == \
Derivative(f(x, y, z), x, y, z, z)
assert Derivative(Derivative(f(x, y, z), x), y)._eval_derivative(z) == \
Derivative(f(x, y, z), x, y, z)
def test_straight_line():
F = f(x)
Fd = F.diff(x)
L = sqrt(1 + Fd**2)
assert diff(L, F) == 0
assert diff(L, Fd) == Fd / sqrt(1 + Fd**2)
