def test_pretty_RootSum():
expr = RootSum(x**5 + 11 * x - 2, auto=False)
ascii_str = \
"""\
/ 5 \\\n\
RootSum\\x + 11*x - 2/\
"""
ucode_str = \
u"""\
⎛ 5 ⎞\n\
RootSum⎝x + 11⋅x - 2⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = RootSum(x**5 + 11 * x - 2, Lambda(z, exp(z)))
ascii_str = \
"""\
/ 5 / z\\\\\n\
RootSum\\x + 11*x - 2, Lambda\\z, e //\
"""
ucode_str = \
u"""\
⎛ 5 ⎛ z⎞⎞\n\
RootSum⎝x + 11⋅x - 2, Λ⎝z, ℯ ⎠⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_RootsOf():
f = Poly((x-4)**4, x)
roots = RootsOf(f)
assert roots.count == 4
assert list(roots.roots()) == [ Integer(4),
Integer(4), Integer(4), Integer(4) ]
assert RootSum(lambda r: r**2, f) == 64
roots = RootsOf(x**5+x+1, x)
assert roots.count == 5
f = Poly(x**5+x+1, x)
assert list(roots.roots()) == [ RootOf(f, 0), RootOf(f, 1),
RootOf(f, 2), RootOf(f, 3), RootOf(f, 4) ]
assert RootSum(lambda r: r**2, f).doit() == RootOf(f, 0)**2 + \
RootOf(f, 1)**2 + RootOf(f, 2)**2 + RootOf(f, 3)**2 + RootOf(f, 4)**2
assert RootSum(Lambda(x, x), Poly(0, x), evaluate=True) == S.Zero
assert RootSum(Lambda(x, x), Poly(0, x), evaluate=False) != S.Zero
assert RootSum(Lambda(x, x), Poly(x-1, x), evaluate=False).doit() == S.One
def test_apart_full():
f = 1 / (x**2 + 1)
assert apart(f, full=False) == f
assert (apart(
f, full=True) == -RootSum(x**2 + 1, Lambda(a, a /
(x - a)), auto=False) / 2)
f = 1 / (x**3 + x + 1)
assert apart(f, full=False) == f
assert apart(f, full=True) == RootSum(
x**3 + x + 1,
Lambda(
a,
(a**2 * Rational(6, 31) - a * Rational(9, 31) + Rational(4, 31)) /
(x - a),
),
auto=False,
)
f = 1 / (x**5 + 1)
assert apart(f, full=False) == (Rational(-1, 5)) * (
(x**3 - 2 * x**2 + 3 * x - 4) /
(x**4 - x**3 + x**2 - x + 1)) + (Rational(1, 5)) / (x + 1)
assert apart(f, full=True) == -RootSum(
x**4 - x**3 + x**2 - x + 1, Lambda(a, a / (x - a)), auto=False) / 5 + (
Rational(1, 5)) / (x + 1)
def test_apart_full():
f = 1 / (x**2 + 1)
assert apart(f, full=False) == f
assert apart(f, full=True) == \
-RootSum(x**2 + 1, Lambda(a, a/(x - a)), auto=False)/2
f = 1 / (x**3 + x + 1)
assert apart(f, full=False) == f
assert apart(f, full=True) == \
RootSum(x**3 + x + 1,
Lambda(a, (6*a**2/31 - 9*a/31 + S(4)/31)/(x - a)), auto=False)
f = 1 / (x**5 + 1)
assert apart(f, full=False) == \
(-S(1)/5)*((x**3 - 2*x**2 + 3*x - 4)/(x**4 - x**3 + x**2 -
x + 1)) + (S(1)/5)/(x + 1)
assert apart(f, full=True) == \
-RootSum(x**4 - x**3 + x**2 - x + 1,
Lambda(a, a/(x - a)), auto=False)/5 + (S(1)/5)/(x + 1)
def test_latex_RootSum():
assert latex(RootSum(x**5 + x + 3, sin)) == \
r"\operatorname{RootSum} {\left(x^{5} + x + 3, \Lambda {\left (x, \sin{\left (x \right )} \right )}\right)}"
def ratint(f, x, **flags):
"""Performs indefinite integration of rational functions.
Given a field :math:`K` and a rational function :math:`f = p/q`,
where :math:`p` and :math:`q` are polynomials in :math:`K[x]`,
returns a function :math:`g` such that :math:`f = g'`.
>>> from sympy.integrals.rationaltools import ratint
>>> from sympy.abc import x
>>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
(12*x + 6)/(x**2 - 1) + 4*log(x - 2) - 4*log(x + 1)
References
==========
.. [Bro05] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70
See Also
========
sympy.integrals.integrals.Integral.doit
ratint_logpart, ratint_ratpart
"""
if type(f) is not tuple:
p, q = f.as_numer_denom()
else:
p, q = f
p, q = Poly(p, x, composite=False, field=True), Poly(q,
x,
composite=False,
field=True)
coeff, p, q = p.cancel(q)
poly, p = p.div(q)
result = poly.integrate(x).as_expr()
if p.is_zero:
return coeff * result
g, h = ratint_ratpart(p, q, x)
P, Q = h.as_numer_denom()
P = Poly(P, x)
Q = Poly(Q, x)
q, r = P.div(Q)
result += g + q.integrate(x).as_expr()
if not r.is_zero:
symbol = flags.get('symbol', 't')
if not isinstance(symbol, Symbol):
t = Dummy(symbol)
else:
t = symbol.as_dummy()
L = ratint_logpart(r, Q, x, t)
real = flags.get('real')
if real is None:
if type(f) is not tuple:
atoms = f.atoms()
else:
p, q = f
atoms = p.atoms() | q.atoms()
for elt in atoms - set([x]):
if not elt.is_real:
real = False
break
else:
real = True
eps = S(0)
if not real:
for h, q in L:
eps += RootSum(q,
Lambda(t, t * log(h.as_expr())),
quadratic=True)
else:
for h, q in L:
R = log_to_real(h, q, x, t)
if R is not None:
eps += R
else:
eps += RootSum(q,
Lambda(t, t * log(h.as_expr())),
quadratic=True)
result += eps
return coeff * result
def ratint(f, x, **flags):
"""Performs indefinite integration of rational functions.
Given a field K and a rational function f = p/q, where p and q
are polynomials in K[x], returns a function g such that f = g'.
>>> from sympy.integrals.rationaltools import ratint
>>> from sympy.abc import x
>>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
-4*log(1 + x) + 4*log(-2 + x) - (6 + 12*x)/(1 - x**2)
References
==========
.. [Bro05] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70
"""
if type(f) is not tuple:
p, q = f.as_numer_denom()
else:
p, q = f
p, q = Poly(p, x), Poly(q, x)
c, p, q = p.cancel(q)
poly, p = p.div(q)
poly = poly.to_field()
result = c * poly.integrate(x).as_basic()
if p.is_zero:
return result
g, h = ratint_ratpart(p, q, x)
P, Q = h.as_numer_denom()
P = Poly(P, x)
Q = Poly(Q, x)
q, r = P.div(Q)
result += g + q.integrate(x).as_basic()
if not r.is_zero:
symbol = flags.get('symbol', 't')
if not isinstance(symbol, Symbol):
t = Dummy(symbol)
else:
t = symbol
L = ratint_logpart(r, Q, x, t)
real = flags.get('real')
if real is None:
if type(f) is not tuple:
atoms = f.atoms()
else:
p, q = f
atoms = p.atoms() \
| q.atoms()
for elt in atoms - set([x]):
if not elt.is_real:
real = False
break
else:
real = True
eps = S(0)
if not real:
for h, q in L:
eps += RootSum(Lambda(t, t * log(h.as_basic())), q)
else:
for h, q in L:
R = log_to_real(h, q, x, t)
if R is not None:
eps += R
else:
eps += RootSum(Lambda(t, t * log(h.as_basic())), q)
result += eps
return result
def test_latex_RootSum():
assert latex(RootSum(x**5 + x + 3, sin)) == \
r"\operatorname{RootSum}\left(x^{5} + x + 3, \operatorname{\Lambda}\left(x, \operatorname{sin}\left(x\right)\right)\right)"
