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_func_deriv():
assert f(x).diff(x) == Derivative(f(x), x)
# issue sympy/sympy#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, y)
assert Derivative(f(x, y), y, x).args[1:] == (y, x)
assert (Derivative(f(x, y), x, y) - Derivative(f(x, y), y, x)).doit() == 0
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_functional_diffgeom_ch3():
x0, y0 = symbols('x0, y0', extended_real=True)
x, y, t = symbols('x, y, t', extended_real=True)
f = Function('f')
b1 = Function('b1')
b2 = Function('b2')
p_r = R2_r.point([x0, y0])
s_field = f(R2.x, R2.y)
v_field = b1(R2.x) * R2.e_x + b2(R2.y) * R2.e_y
assert v_field.rcall(s_field).rcall(p_r).doit() == b1(x0) * Derivative(
f(x0, y0), x0) + b2(y0) * Derivative(f(x0, y0), y0)
assert R2.e_x(R2.r**2).rcall(p_r) == 2 * x0
v = R2.e_x + 2 * R2.e_y
s = R2.r**2 + 3 * R2.x
assert v.rcall(s).rcall(p_r).doit() == 2 * x0 + 4 * y0 + 3
circ = -R2.y * R2.e_x + R2.x * R2.e_y
series = intcurve_series(circ, t, R2_r.point([1, 0]), coeffs=True)
series_x, series_y = zip(*series)
assert all(term == cos(t).taylor_term(i, t)
for i, term in enumerate(series_x))
assert all(term == sin(t).taylor_term(i, t)
for i, term in enumerate(series_y))
def test_sympyissue_5183():
assert abs(x + x**2).series(n=1) == O(x)
assert abs(x + x**2).series(n=2) == x + O(x**2)
assert ((1 + x)**2).series(x, n=6) == 1 + 2 * x + x**2
assert (1 + 1 / x).series() == 1 + 1 / x
assert Derivative(exp(x).series(), x).doit() == \
1 + x + x**2/2 + x**3/6 + x**4/24 + Derivative(O(x**6), x)
def test_deriv2():
assert (x**3).diff(x) == 3 * x**2
assert (x**3).diff(x, evaluate=False) != 3 * x**2
assert (x**3).diff(x, evaluate=False) == Derivative(x**3, x)
assert diff(x**3, x) == 3 * x**2
assert diff(x**3, x, evaluate=False) != 3 * x**2
assert diff(x**3, x, evaluate=False) == Derivative(x**3, x)
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)
pytest.raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
pytest.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)) == Subs(Derivative(f(x), x),
(x, ), (g(x), ))
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_diofantissue_376():
f = symbols('f', cls=Function)
e1 = Subs(Derivative(f(x), x), (x, y * z))
e2 = Subs(Derivative(f(t), t), (t, y * z))
assert (x * e1).subs({e2: y}) == x * y
e3 = Subs(Derivative(f(t), t), (t, y * z**2))
assert e3.subs({e2: y}) == e3
e4 = Subs(Derivative(f(t), t, t), (t, y * z))
assert e4.subs({e2: y}) == e4
def test_derivative_bug1():
f = Function("f")
a = Wild("a", exclude=[f, x])
b = Wild("b", exclude=[f])
pattern = a * Derivative(f(x), x, x) + b
expr = Derivative(f(x), x) + x**2
d1 = {b: x**2}
d2 = expr.match(pattern.xreplace(d1))
assert d2 is None
def test_derivative_bug1():
f = Function('f')
a = Wild('a', exclude=[f, x])
b = Wild('b', exclude=[f])
pattern = a * Derivative(f(x), x, x) + b
expr = Derivative(f(x), x) + x**2
d1 = {b: x**2}
d2 = expr.match(pattern.xreplace(d1))
assert d2 is None
def test_unhandled():
class MyExpr(Expr):
def _eval_derivative(self, s):
if not s.name.startswith('xi'):
return self
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_derivative_bug1():
f = Function("f")
a = Wild("a", exclude=[f, x])
b = Wild("b", exclude=[f])
pattern = a * Derivative(f(x), x, x) + b
expr = Derivative(f(x), x) + x**2
d1 = {b: x**2}
d2 = pattern.xreplace(d1).matches(expr, d1)
assert d2 is None
def test_vector_diff_integrate():
f = Function('f')
v = f(a)*C.i + a**2*C.j - C.k
assert Derivative(v, a) == Derivative((f(a))*C.i +
a**2*C.j + (-1)*C.k, a)
assert (diff(v, a) == v.diff(a) == Derivative(v, a).doit() ==
(Derivative(f(a), a))*C.i + 2*a*C.j)
assert (Integral(v, a) == (Integral(f(a), a))*C.i +
(Integral(a**2, a))*C.j + (Integral(-1, a))*C.k)
def test_diofantissue_376():
x, y, z, t = symbols('x y z t')
f = symbols('f', cls=Function)
e1 = Subs(Derivative(f(x), x), x, y * z)
e2 = Subs(Derivative(f(t), t), t, y * z)
assert (x * e1).subs(e2, y) == x * y
e3 = Subs(Derivative(f(t), t), t, y * z**2)
assert e3.subs(e2, y) == e3
e4 = Subs(Derivative(f(t), t, t), t, y * z)
assert e4.subs(e2, y) == e4
def test_simultaneous_subs():
reps = {x: 0, y: 0}
assert (x / y).subs(reps) != (y / x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
reps = reps.items()
assert (x / y).subs(reps) != (y / x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
assert Derivative(x, y, z).subs(reps, simultaneous=True) == \
Subs(Derivative(0, y, z), (y, 0))
def test_derivative1():
p, q = map(Wild, 'pq')
f = Function('f', nargs=1)
fd = Derivative(f(x), x)
assert fd.match(p) == {p: fd}
assert (fd + 1).match(p + 1) == {p: fd}
assert fd.match(fd) == {}
assert (3 * fd).match(p * fd) is not None
assert (3 * fd - 1).match(p * fd + q) == {p: 3, q: -1}
def test_derivative1():
p, q = map(Wild, 'pq')
f = Function('f', nargs=1)
fd = Derivative(f(x), x)
assert fd.match(p) == {p: fd}
assert (fd + 1).match(p + 1) == {p: fd}
assert fd.match(fd) == {}
assert (3*fd).match(p*fd) is not None
assert (3*fd - 1).match(p*fd + q) == {p: 3, q: -1}
def test_sympyissue_7231():
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_simple_1():
assert x.series(x, n=5) == x
assert y.series(x, n=5) == y
assert (1/(x*y)).series(y, n=5) == 1/(x*y)
assert Rational(3, 4).series(x, n=5) == Rational(3, 4)
assert x.series(x) == x
# issue sympy/sympy#5183
assert ((1 + x)**2).series(x) == 1 + 2*x + x**2
assert (1 + 1/x).series() == 1 + 1/x
assert (Derivative(exp(x).series(), x).doit() ==
1 + x + x**2/2 + x**3/6 + x**4/24 + Derivative(O(x**6), x))
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 sympy/sympy#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_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_derivatives_sympyissue_4757():
x = Symbol('x', extended_real=True)
y = Symbol('y', imaginary=True)
f = Function('f')
assert re(f(x)).diff(x) == re(f(x).diff(x))
assert im(f(x)).diff(x) == im(f(x).diff(x))
assert re(f(y)).diff(y) == -I*im(f(y).diff(y))
assert im(f(y)).diff(y) == -I*re(f(y).diff(y))
assert re(f(z)).diff(z) == Derivative(re(f(z)), z)
assert im(f(z)).diff(z) == Derivative(im(f(z)), z)
assert Abs(f(x)).diff(x).subs(f(x), 1 + I*x).doit() == x/sqrt(1 + x**2)
assert arg(f(x)).diff(x).subs(f(x), 1 + I*x**2).doit() == 2*x/(1 + x**4)
assert Abs(f(y)).diff(y).subs(f(y), 1 + y).doit() == -y/sqrt(1 - y**2)
assert arg(f(y)).diff(y).subs(f(y), I + y**2).doit() == 2*y/(1 + y**4)
def test_sympyissue_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) == 0
e5 = Subs(Derivative(f(x), x), (y, y), (z, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Subs(Derivative(f(y), y, y), (y, g(x)))
def test_conjugate():
a = Symbol('a', extended_real=True)
b = Symbol('b', imaginary=True)
assert conjugate(a) == a
assert conjugate(I * a) == -I * a
assert conjugate(b) == -b
assert conjugate(I * b) == I * b
assert conjugate(a * b) == -a * b
assert conjugate(I * a * b) == I * a * b
x = Symbol('x')
y = Symbol('y')
assert conjugate(conjugate(x)) == x
assert conjugate(x + y) == conjugate(x) + conjugate(y)
assert conjugate(x - y) == conjugate(x) - conjugate(y)
assert conjugate(x * y) == conjugate(x) * conjugate(y)
assert conjugate(x / y) == conjugate(x) / conjugate(y)
assert conjugate(-x) == -conjugate(x)
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert conjugate(z).diff(z) == Derivative(conjugate(z), z)
# issue sympy/sympy#4754
x = Symbol('x', extended_real=True)
y = Symbol('y', imaginary=True)
f = Function('f')
assert (f(x).conjugate()).diff(x) == (f(x).diff(x)).conjugate()
assert (f(y).conjugate()).diff(y) == -(f(y).diff(y)).conjugate()
def test_simple():
# issue sympy/sympy#5223
assert Integer(1).series(x) == 1
pytest.raises(ValueError, lambda: cos(x + y).series())
pytest.raises(ValueError, lambda: x.series(dir=''))
pytest.raises(ValueError, lambda: x.series(dir=0))
assert Derivative(x**2 + x**3 * y**2, (x, 2),
(y, 1)).series(x).simplify() == 12 * x * y + O(x**6)
assert (1 + x).getn() is None
# issue sympy/sympy#8805
assert Integer(1).series(n=8) == 1
# issue sympy/sympy#5223
assert (cos(x).series(x, 1) -
cos(x + 1).series(x).subs({x: x - 1})).removeO() == 0
assert abs(x).series(x, oo, n=5, dir=-1) == x
assert abs(x).series(x, -oo, n=5, dir=+1) == -x
assert abs(-x).series(x, oo, n=5, dir=-1) == x
assert abs(-x).series(x, -oo, n=5, dir=+1) == -x
# issue sympy/sympy#7203
assert cos(x).series(x, pi, 3) == -1 + (x - pi)**2 / 2 + O((x - pi)**3,
(x, pi))
def test_hyper():
pytest.raises(TypeError, lambda: hyper(1, 2, z))
assert hyper((1, 2), (1, ), z) == hyper(Tuple(1, 2), Tuple(1), z)
h = hyper((1, 2), (3, 4, 5), z)
assert h.ap == Tuple(1, 2)
assert h.bq == Tuple(3, 4, 5)
assert h.argument == z
assert h.is_commutative is True
# just a few checks to make sure that all arguments go where they should
assert tn(hyper(Tuple(), Tuple(), z), exp(z), z)
assert tn(z * hyper((1, 1), Tuple(2), -z), log(1 + z), z)
# differentiation
h = hyper((randcplx(), randcplx(), randcplx()), (randcplx(), randcplx()),
z)
assert td(h, z)
a1, a2, b1, b2, b3 = symbols('a1:3, b1:4')
assert hyper((a1, a2), (b1, b2, b3), z).diff(z) == \
a1*a2/(b1*b2*b3) * hyper((a1 + 1, a2 + 1), (b1 + 1, b2 + 1, b3 + 1), z)
# differentiation wrt parameters is not supported
assert hyper([z], [], z).diff(z) == Derivative(hyper([z], [], z), z)
# hyper is unbranched wrt parameters
assert hyper([polar_lift(z)], [polar_lift(k)], polar_lift(x)) == \
hyper([z], [k], polar_lift(x))
assert hyper((1, 2, 3), [3, 4], 1).is_number
assert not hyper((1, 2, 3), [3, x], 1).is_number
def test_derivative_by_array():
bexpr = x * y**2 * exp(z) * log(t)
sexpr = sin(bexpr)
cexpr = cos(bexpr)
a = Array([sexpr])
assert derive_by_array(
sexpr, t) == x * y**2 * exp(z) * cos(x * y**2 * exp(z) * log(t)) / t
assert derive_by_array(sexpr, [x, y, z]) == Array(
[bexpr / x * cexpr, 2 * y * bexpr / y**2 * cexpr, bexpr * cexpr])
assert derive_by_array(a,
[x, y, z]) == Array([[bexpr / x * cexpr],
[2 * y * bexpr / y**2 * cexpr],
[bexpr * cexpr]])
assert derive_by_array(sexpr, [[x, y], [z, t]]) == Array(
[[bexpr / x * cexpr, 2 * y * bexpr / y**2 * cexpr],
[bexpr * cexpr, bexpr / log(t) / t * cexpr]])
assert derive_by_array(a, [[x, y], [z, t]]) == Array(
[[[bexpr / x * cexpr], [2 * y * bexpr / y**2 * cexpr]],
[[bexpr * cexpr], [bexpr / log(t) / t * cexpr]]])
assert derive_by_array([[x, y], [z, t]], [x, y]) == Array([[[1, 0], [0,
0]],
[[0, 1], [0,
0]]])
assert derive_by_array([[x, y], [z, t]],
[[x, y], [z, t]]) == Array([[[[1, 0], [0, 0]],
[[0, 1], [0, 0]]],
[[[0, 0], [1, 0]],
[[0, 0], [0, 1]]]])
pytest.raises(ValueError,
lambda: derive_by_array(x, [Derivative(x**2, x)]))
def test_Derivative_bug1():
f = Function("f")
x = abc.x
a = Wild("a", exclude=[f(x)])
b = Wild("b", exclude=[f(x)])
eq = f(x).diff(x)
assert eq.match(a * Derivative(f(x), x) + b) == {a: 1, b: 0}
def test_jacobi():
assert jacobi(0, a, b, x) == 1
assert jacobi(1, a, b, x) == a / 2 - b / 2 + x * (a / 2 + b / 2 + 1)
assert (jacobi(2, a, b, x) == a**2 / 8 - a * b / 4 - a / 8 + b**2 / 8 -
b / 8 + x**2 * (a**2 / 8 + a * b / 4 + 7 * a / 8 + b**2 / 8 +
7 * b / 8 + Rational(3, 2)) + x *
(a**2 / 4 + 3 * a / 4 - b**2 / 4 - 3 * b / 4) - S.Half)
assert jacobi(n, a, a, x) == RisingFactorial(a + 1, n) * gegenbauer(
n, a + Rational(1, 2), x) / RisingFactorial(2 * a + 1, n)
assert jacobi(n, a, -a,
x) == ((-1)**a * (-x + 1)**(-a / 2) * (x + 1)**(a / 2) *
assoc_legendre(n, a, x) * factorial(-a + n) *
gamma(a + n + 1) / (factorial(a + n) * gamma(n + 1)))
assert jacobi(n, -b, b, x) == ((-x + 1)**(b / 2) * (x + 1)**(-b / 2) *
assoc_legendre(n, b, x) *
gamma(-b + n + 1) / gamma(n + 1))
assert jacobi(n, 0, 0, x) == legendre(n, x)
assert jacobi(n, S.Half, S.Half, x) == RisingFactorial(Rational(
3, 2), n) * chebyshevu(n, x) / factorial(n + 1)
assert jacobi(n, -S.Half, -S.Half, x) == RisingFactorial(
Rational(1, 2), n) * chebyshevt(n, x) / factorial(n)
X = jacobi(n, a, b, x)
assert isinstance(X, jacobi)
assert jacobi(n, a, b, -x) == (-1)**n * jacobi(n, b, a, x)
assert jacobi(n, a, b, 0) == 2**(-n) * gamma(a + n + 1) * hyper(
(-b - n, -n), (a + 1, ), -1) / (factorial(n) * gamma(a + 1))
assert jacobi(n, a, b, 1) == RisingFactorial(a + 1, n) / factorial(n)
m = Symbol("m", positive=True)
assert jacobi(m, a, b, oo) == oo * RisingFactorial(a + b + m + 1, m)
assert conjugate(jacobi(m, a, b, x)) == \
jacobi(m, conjugate(a), conjugate(b), conjugate(x))
assert diff(jacobi(n, a, b, x), n) == Derivative(jacobi(n, a, b, x), n)
assert diff(jacobi(n, a, b, x), x) == \
(a/2 + b/2 + n/2 + Rational(1, 2))*jacobi(n - 1, a + 1, b + 1, x)
# XXX see issue sympy/sympy#5539
assert str(jacobi(n, a, b, x).diff(a)) == \
("Sum((jacobi(n, a, b, x) + (a + b + 2*_k + 1)*RisingFactorial(b + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)))/(a + b + n + _k + 1), (_k, 0, n - 1))")
assert str(jacobi(n, a, b, x).diff(b)) == \
("Sum(((-1)**(n - _k)*(a + b + 2*_k + 1)*RisingFactorial(a + "
"_k + 1, n - _k)*jacobi(_k, a, b, x)/((n - _k)*RisingFactorial(a + "
"b + _k + 1, n - _k)) + jacobi(n, a, b, x))/(a + b + n + "
"_k + 1), (_k, 0, n - 1))")
assert jacobi_normalized(n, a, b, x) == \
(jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)
/ ((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))))
pytest.raises(ValueError, lambda: jacobi(-2.1, a, b, x))
pytest.raises(ValueError,
lambda: jacobi(Dummy(positive=True, integer=True), 1, 2, oo))
pytest.raises(ArgumentIndexError, lambda: jacobi(n, a, b, x).fdiff(5))
def test_derivatives():
assert zeta(x, a).diff(x) == Derivative(zeta(x, a), x)
assert zeta(x, a).diff(a) == -x * zeta(x + 1, a)
assert zeta(z).diff(z) == Derivative(zeta(z), z)
assert lerchphi(
z, s, a).diff(z) == (lerchphi(z, s - 1, a) - a * lerchphi(z, s, a)) / z
pytest.raises(ArgumentIndexError, lambda: lerchphi(z, s, a).fdiff(4))
assert lerchphi(z, s, a).diff(a) == -s * lerchphi(z, s + 1, a)
assert polylog(s, z).diff(z) == polylog(s - 1, z) / z
pytest.raises(ArgumentIndexError, lambda: polylog(s, z).fdiff(3))
b = randcplx()
c = randcplx()
assert td(zeta(b, x), x)
assert td(polylog(b, z), z)
assert td(lerchphi(c, b, x), x)
assert td(lerchphi(x, b, c), x)
def test_eval_derivative():
assert Sum(x, (x, 1, 2)).diff(x) == 0
assert Sum(x * y, (x, 1, 2)).diff(x) == 0
assert Sum(x * y, (y, 1, 2)).diff(x) == Sum(y, (y, 1, 2))
e = Sum(x * y, (x, 1, a))
assert e.diff(a) == Derivative(e, a)
assert Sum(x*y, (x, 1, 3), (a, 2, 5)).diff(y) == \
Sum(x*y, (x, 1, 3), (a, 2, 5)).doit().diff(y) == 24
def test_components():
assert components(x * y, x) == {x}
assert components(1 / (x + y), x) == {x}
assert components(sin(x), x) == {sin(x), x}
assert components(sin(x)*sqrt(log(x)), x) == \
{log(x), sin(x), sqrt(log(x)), x}
assert components(x*sin(exp(x)*y), x) == \
{sin(y*exp(x)), x, exp(x)}
assert components(x**Rational(17, 54)/sqrt(sin(x)), x) == \
{sin(x), root(x, 54), sqrt(sin(x)), x}
assert components(f(x), x) == \
{x, f(x)}
assert components(Derivative(f(x), x), x) == \
{x, f(x), Derivative(f(x), x)}
assert components(f(x)*diff(f(x), x), x) == \
{x, f(x), Derivative(f(x), x), Derivative(f(x), x)}
def test_derivative2():
f = Function("f")
a = Wild("a", exclude=[f, x])
b = Wild("b", exclude=[f])
e = Derivative(f(x), x)
assert e.match(Derivative(f(x), x)) == {}
assert e.match(Derivative(f(x), x, x)) is None
e = Derivative(f(x), x, x)
assert e.match(Derivative(f(x), x)) is None
assert e.match(Derivative(f(x), x, x)) == {}
e = Derivative(f(x), x) + x**2
assert e.match(a*Derivative(f(x), x) + b) == {a: 1, b: x**2}
assert e.match(a*Derivative(f(x), x, x) + b) is None
e = Derivative(f(x), x, x) + x**2
assert e.match(a*Derivative(f(x), x) + b) is None
assert e.match(a*Derivative(f(x), x, x) + b) == {a: 1, b: x**2}
def test_sympyissue_11313():
# test Derivative series & as_leading_term
assert Derivative(x**3 + x**4, x).as_leading_term(x).doit() == 3*x**2
s = Derivative(sin(x), x).series(x, n=3)
assert s == Derivative(-x**3/6, x) + Derivative(x, x) + O(x**3)
assert s.doit() == 1 - x**2/2 + O(x**3)
def test_derivative_subs3():
x = Symbol('x')
dex = Derivative(exp(x), x)
assert Derivative(dex, x).subs({dex: exp(x)}) == dex
assert dex.subs({exp(x): dex}) == Derivative(exp(x), x, x)
def test_deriv_sub_bug3():
y = Symbol('y')
f = Function('f')
pat = Derivative(f(x), x, x)
assert pat.subs({y: y**2}) == Derivative(f(x), x, x)
assert pat.subs({y: y**2}) != Derivative(f(x), x)
def test_doit():
n = Symbol('n', integer=True)
f = Sum(2 * n * x, (n, 1, 3))
d = Derivative(f, x)
assert d.doit() == 12
assert d.doit(deep=False) == Sum(2*n, (n, 1, 3))
