def ratint_ratpart(f, g, x):
"""Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
"""
f = Poly(f, x)
g = Poly(g, x)
u, v, _ = g.cofactors(g.diff())
n = u.degree()
m = v.degree()
A_coeffs = [ Dummy('a' + str(n-i)) for i in xrange(0, n) ]
B_coeffs = [ Dummy('b' + str(m-i)) for i in xrange(0, m) ]
C_coeffs = A_coeffs + B_coeffs
A = Poly(A_coeffs, x, domain=ZZ[C_coeffs])
B = Poly(B_coeffs, x, domain=ZZ[C_coeffs])
H = f - A.diff()*v + A*(u.diff()*v).quo(u) - B*u
result = solve(H.coeffs(), C_coeffs)
A = A.as_expr().subs(result)
B = B.as_expr().subs(result)
rat_part = cancel(A/u.as_expr(), x)
log_part = cancel(B/v.as_expr(), x)
return rat_part, log_part
def ratint_ratpart(f, g, x):
"""
Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
Examples
========
>>> from sympy.integrals.rationaltools import ratint_ratpart
>>> from sympy.abc import x, y
>>> from sympy import Poly
>>> ratint_ratpart(Poly(1, x, domain='ZZ'),
... Poly(x + 1, x, domain='ZZ'), x)
(0, 1/(x + 1))
>>> ratint_ratpart(Poly(1, x, domain='EX'),
... Poly(x**2 + y**2, x, domain='EX'), x)
(0, 1/(x**2 + y**2))
>>> ratint_ratpart(Poly(36, x, domain='ZZ'),
... Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x, domain='ZZ'), x)
((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2))
See Also
========
ratint, ratint_logpart
"""
from sympy import solve
f = Poly(f, x)
g = Poly(g, x)
u, v, _ = g.cofactors(g.diff())
n = u.degree()
m = v.degree()
A_coeffs = [ Dummy('a' + str(n - i)) for i in range(0, n) ]
B_coeffs = [ Dummy('b' + str(m - i)) for i in range(0, m) ]
C_coeffs = A_coeffs + B_coeffs
A = Poly(A_coeffs, x, domain=ZZ[C_coeffs])
B = Poly(B_coeffs, x, domain=ZZ[C_coeffs])
H = f - A.diff()*v + A*(u.diff()*v).quo(u) - B*u
result = solve(H.coeffs(), C_coeffs)
A = A.as_expr().subs(result)
B = B.as_expr().subs(result)
rat_part = cancel(A/u.as_expr(), x)
log_part = cancel(B/v.as_expr(), x)
return rat_part, log_part
def _solve_as_poly(f, symbol, solveset_solver, invert_func):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f, symbol, cubics=True, quartics=True,
quintics=True, domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
else:
poly = Poly(f)
if poly is None:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True,
quintics=True).keys())
if len(poly_solns) < deg:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
if gen != symbol:
y = Dummy('y')
lhs, rhs_s = invert_func(gen, y, symbol)
if lhs is symbol:
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with free
# variables because that makes the solution more complicated. For
# example expand_complex(a) returns re(a) + I*im(a)
if all([s.free_symbols == set() and not isinstance(s, RootOf)
for s in result]):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
return result
else:
return ConditionSet(symbol, Eq(f, 0), S.Complexes)
def apart_undetermined_coeffs(P, Q):
"""Partial fractions via method of undetermined coefficients. """
X = numbered_symbols(cls=Dummy)
partial, symbols = [], []
_, factors = Q.factor_list()
for f, k in factors:
n, q = f.degree(), Q
for i in xrange(1, k + 1):
coeffs, q = take(X, n), q.quo(f)
partial.append((coeffs, q, f, i))
symbols.extend(coeffs)
dom = Q.get_domain().inject(*symbols)
F = Poly(0, Q.gen, domain=dom)
for i, (coeffs, q, f, k) in enumerate(partial):
h = Poly(coeffs, Q.gen, domain=dom)
partial[i] = (h, f, k)
q = q.set_domain(dom)
F += h * q
system, result = [], S(0)
for (k,), coeff in F.terms():
system.append(coeff - P.nth(k))
from sympy.solvers import solve
solution = solve(system, symbols)
for h, f, k in partial:
h = h.as_expr().subs(solution)
result += h / f.as_expr() ** k
return result
def gosper_term(f, n):
r"""
Compute Gosper's hypergeometric term for ``f``.
Suppose ``f`` is a hypergeometric term such that:
.. math::
s_n = \sum_{k=0}^{n-1} f_k
and `f_k` doesn't depend on `n`. Returns a hypergeometric
term `g_n` such that `g_{n+1} - g_n = f_n`.
**Examples**
>>> from sympy.concrete.gosper import gosper_term
>>> from sympy.functions import factorial
>>> from sympy.abc import n
>>> gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n)
(-n - 1/2)/(n + 1/4)
"""
r = hypersimp(f, n)
if r is None:
return None # 'f' is *not* a hypergeometric term
p, q = r.as_numer_denom()
A, B, C = gosper_normal(p, q, n)
B = B.shift(-1)
N = S(A.degree())
M = S(B.degree())
K = S(C.degree())
if (N != M) or (A.LC() != B.LC()):
D = set([K - max(N, M)])
elif not N:
D = set([K - N + 1, S(0)])
else:
D = set([K - N + 1, (B.nth(N - 1) - A.nth(N - 1)) / A.LC()])
for d in set(D):
if not d.is_Integer or d < 0:
D.remove(d)
if not D:
return None # 'f(n)' is *not* Gosper-summable
d = max(D)
coeffs = symbols("c:%s" % (d + 1), cls=Dummy)
domain = A.get_domain().inject(*coeffs)
x = Poly(coeffs, n, domain=domain)
H = A * x.shift(1) - B * x - C
solution = solve(H.coeffs(), coeffs)
if solution is None:
return None # 'f(n)' is *not* Gosper-summable
x = x.as_expr().subs(solution)
for coeff in coeffs:
if coeff not in solution:
x = x.subs(coeff, 0)
if x is S.Zero:
return None # 'f(n)' is *not* Gosper-summable
else:
return B.as_expr() * x / C.as_expr()
def _solve_inequality(ie, s, linear=False):
"""Return the inequality with s isolated on the left, if possible.
If the relationship is non-linear, a solution involving And or Or
may be returned. False or True are returned if the relationship
is never True or always True, respectively.
If `linear` is True (default is False) an `s`-dependent expression
will be isoloated on the left, if possible
but it will not be solved for `s` unless the expression is linear
in `s`. Furthermore, only "safe" operations which don't change the
sense of the relationship are applied: no division by an unsigned
value is attempted unless the relationship involves Eq or Ne and
no division by a value not known to be nonzero is ever attempted.
Examples
========
>>> from sympy import Eq, Symbol
>>> from sympy.solvers.inequalities import _solve_inequality as f
>>> from sympy.abc import x, y
For linear expressions, the symbol can be isolated:
>>> f(x - 2 < 0, x)
x < 2
>>> f(-x - 6 < x, x)
x > -3
Sometimes nonlinear relationships will be False
>>> f(x**2 + 4 < 0, x)
False
Or they may involve more than one region of values:
>>> f(x**2 - 4 < 0, x)
(-2 < x) & (x < 2)
To restrict the solution to a relational, set linear=True
and only the x-dependent portion will be isolated on the left:
>>> f(x**2 - 4 < 0, x, linear=True)
x**2 < 4
Division of only nonzero quantities is allowed, so x cannot
be isolated by dividing by y:
>>> y.is_nonzero is None # it is unknown whether it is 0 or not
True
>>> f(x*y < 1, x)
x*y < 1
And while an equality (or unequality) still holds after dividing by a
non-zero quantity
>>> nz = Symbol('nz', nonzero=True)
>>> f(Eq(x*nz, 1), x)
Eq(x, 1/nz)
the sign must be known for other inequalities involving > or <:
>>> f(x*nz <= 1, x)
nz*x <= 1
>>> p = Symbol('p', positive=True)
>>> f(x*p <= 1, x)
x <= 1/p
When there are denominators in the original expression that
are removed by expansion, conditions for them will be returned
as part of the result:
>>> f(x < x*(2/x - 1), x)
(x < 1) & Ne(x, 0)
"""
from sympy.solvers.solvers import denoms
if s not in ie.free_symbols:
return ie
if ie.rhs == s:
ie = ie.reversed
if ie.lhs == s and s not in ie.rhs.free_symbols:
return ie
expr = ie.lhs - ie.rhs
rv = None
try:
p = Poly(expr, s)
if p.degree() == 0:
rv = ie.func(p.as_expr(), 0)
elif not linear and p.degree() > 1:
# handle in except clause
raise NotImplementedError
except (PolynomialError, NotImplementedError):
if not linear:
try:
return reduce_rational_inequalities([[ie]], s)
except PolynomialError:
return solve_univariate_inequality(ie, s)
else:
p = Poly(expr)
e = expanded = p.as_expr() # this is in exanded form
if rv is None:
# Do a safe inversion of e, moving non-s terms
# to the rhs and dividing by a nonzero factor if
# the relational is Eq/Ne; for other relationals
# the sign must also be positive or negative
rhs = 0
b, ax = e.as_independent(s, as_Add=True)
e -= b
rhs -= b
ef = factor_terms(e)
a, e = ef.as_independent(s, as_Add=False)
if (a.is_zero != False or # don't divide by potential 0
a.is_negative ==
a.is_positive == None and # if sign is not known then
ie.rel_op not in ('!=', '==')): # reject if not Eq/Ne
e = ef
a = S.One
rhs /= a
if a.is_positive:
rv = ie.func(e, rhs)
else:
rv = ie.reversed.func(e, rhs)
# return conditions under which the value is
# valid, too.
conds = [rv]
beginning_denoms = denoms(ie.lhs) | denoms(ie.rhs)
current_denoms = denoms(expanded)
for d in beginning_denoms - current_denoms:
conds.append(_solve_inequality(Ne(d, 0), s, linear=linear))
return And(*conds)
def _solve_inequality(ie, s, linear=False):
"""Return the inequality with s isolated on the left, if possible.
If the relationship is non-linear, a solution involving And or Or
may be returned. False or True are returned if the relationship
is never True or always True, respectively.
If `linear` is True (default is False) an `s`-dependent expression
will be isolated on the left, if possible
but it will not be solved for `s` unless the expression is linear
in `s`. Furthermore, only "safe" operations which don't change the
sense of the relationship are applied: no division by an unsigned
value is attempted unless the relationship involves Eq or Ne and
no division by a value not known to be nonzero is ever attempted.
Examples
========
>>> from sympy import Eq, Symbol
>>> from sympy.solvers.inequalities import _solve_inequality as f
>>> from sympy.abc import x, y
For linear expressions, the symbol can be isolated:
>>> f(x - 2 < 0, x)
x < 2
>>> f(-x - 6 < x, x)
x > -3
Sometimes nonlinear relationships will be False
>>> f(x**2 + 4 < 0, x)
False
Or they may involve more than one region of values:
>>> f(x**2 - 4 < 0, x)
(-2 < x) & (x < 2)
To restrict the solution to a relational, set linear=True
and only the x-dependent portion will be isolated on the left:
>>> f(x**2 - 4 < 0, x, linear=True)
x**2 < 4
Division of only nonzero quantities is allowed, so x cannot
be isolated by dividing by y:
>>> y.is_nonzero is None # it is unknown whether it is 0 or not
True
>>> f(x*y < 1, x)
x*y < 1
And while an equality (or inequality) still holds after dividing by a
non-zero quantity
>>> nz = Symbol('nz', nonzero=True)
>>> f(Eq(x*nz, 1), x)
Eq(x, 1/nz)
the sign must be known for other inequalities involving > or <:
>>> f(x*nz <= 1, x)
nz*x <= 1
>>> p = Symbol('p', positive=True)
>>> f(x*p <= 1, x)
x <= 1/p
When there are denominators in the original expression that
are removed by expansion, conditions for them will be returned
as part of the result:
>>> f(x < x*(2/x - 1), x)
(x < 1) & Ne(x, 0)
"""
from sympy.solvers.solvers import denoms
if s not in ie.free_symbols:
return ie
if ie.rhs == s:
ie = ie.reversed
if ie.lhs == s and s not in ie.rhs.free_symbols:
return ie
def classify(ie, s, i):
# return True or False if ie evaluates when substituting s with
# i else None (if unevaluated) or NaN (when there is an error
# in evaluating)
try:
v = ie.subs(s, i)
if v is S.NaN:
return v
elif v not in (True, False):
return
return v
except TypeError:
return S.NaN
rv = None
oo = S.Infinity
expr = ie.lhs - ie.rhs
try:
p = Poly(expr, s)
if p.degree() == 0:
rv = ie.func(p.as_expr(), 0)
elif not linear and p.degree() > 1:
# handle in except clause
raise NotImplementedError
except (PolynomialError, NotImplementedError):
if not linear:
try:
rv = reduce_rational_inequalities([[ie]], s)
except PolynomialError:
rv = solve_univariate_inequality(ie, s)
# remove restrictions wrt +/-oo that may have been
# applied when using sets to simplify the relationship
okoo = classify(ie, s, oo)
if okoo is S.true and classify(rv, s, oo) is S.false:
rv = rv.subs(s < oo, True)
oknoo = classify(ie, s, -oo)
if (oknoo is S.true and classify(rv, s, -oo) is S.false):
rv = rv.subs(-oo < s, True)
rv = rv.subs(s > -oo, True)
if rv is S.true:
rv = (s <= oo) if okoo is S.true else (s < oo)
if oknoo is not S.true:
rv = And(-oo < s, rv)
else:
p = Poly(expr)
conds = []
if rv is None:
e = p.as_expr() # this is in expanded form
# Do a safe inversion of e, moving non-s terms
# to the rhs and dividing by a nonzero factor if
# the relational is Eq/Ne; for other relationals
# the sign must also be positive or negative
rhs = 0
b, ax = e.as_independent(s, as_Add=True)
e -= b
rhs -= b
ef = factor_terms(e)
a, e = ef.as_independent(s, as_Add=False)
if (a.is_zero != False or # don't divide by potential 0
a.is_negative == a.is_positive is None
and # if sign is not known then
ie.rel_op not in ('!=', '==')): # reject if not Eq/Ne
e = ef
a = S.One
rhs /= a
if a.is_positive:
rv = ie.func(e, rhs)
else:
rv = ie.reversed.func(e, rhs)
# return conditions under which the value is
# valid, too.
beginning_denoms = denoms(ie.lhs) | denoms(ie.rhs)
current_denoms = denoms(rv)
for d in beginning_denoms - current_denoms:
c = _solve_inequality(Eq(d, 0), s, linear=linear)
if isinstance(c, Eq) and c.lhs == s:
if classify(rv, s, c.rhs) is S.true:
# rv is permitting this value but it shouldn't
conds.append(~c)
for i in (-oo, oo):
if (classify(rv, s, i) is S.true
and classify(ie, s, i) is not S.true):
conds.append(s < i if i is oo else i < s)
conds.append(rv)
return And(*conds)
def gosper_term(f, n):
r"""
Compute Gosper's hypergeometric term for ``f``.
Suppose ``f`` is a hypergeometric term such that:
.. math::
s_n = \sum_{k=0}^{n-1} f_k
and `f_k` doesn't depend on `n`. Returns a hypergeometric
term `g_n` such that `g_{n+1} - g_n = f_n`.
Examples
========
>>> from sympy.concrete.gosper import gosper_term
>>> from sympy.functions import factorial
>>> from sympy.abc import n
>>> gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n)
(-n - 1/2)/(n + 1/4)
"""
r = hypersimp(f, n)
if r is None:
return None # 'f' is *not* a hypergeometric term
p, q = r.as_numer_denom()
A, B, C = gosper_normal(p, q, n)
B = B.shift(-1)
N = S(A.degree())
M = S(B.degree())
K = S(C.degree())
if (N != M) or (A.LC() != B.LC()):
D = set([K - max(N, M)])
elif not N:
D = set([K - N + 1, S(0)])
else:
D = set([K - N + 1, (B.nth(N - 1) - A.nth(N - 1))/A.LC()])
for d in set(D):
if not d.is_Integer or d < 0:
D.remove(d)
if not D:
return None # 'f(n)' is *not* Gosper-summable
d = max(D)
coeffs = symbols('c:%s' % (d + 1), cls=Dummy)
domain = A.get_domain().inject(*coeffs)
x = Poly(coeffs, n, domain=domain)
H = A*x.shift(1) - B*x - C
solution = solve(H.coeffs(), coeffs)
if solution is None:
return None # 'f(n)' is *not* Gosper-summable
x = x.as_expr().subs(solution)
for coeff in coeffs:
if coeff not in solution:
x = x.subs(coeff, 0)
if x is S.Zero:
return None # 'f(n)' is *not* Gosper-summable
else:
return B.as_expr()*x/C.as_expr()
def ratint_ratpart(f, g, x):
"""
Horowitz-Ostrogradsky algorithm.
Explanation
===========
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
Examples
========
>>> from sympy.integrals.rationaltools import ratint_ratpart
>>> from sympy.abc import x, y
>>> from sympy import Poly
>>> ratint_ratpart(Poly(1, x, domain='ZZ'),
... Poly(x + 1, x, domain='ZZ'), x)
(0, 1/(x + 1))
>>> ratint_ratpart(Poly(1, x, domain='EX'),
... Poly(x**2 + y**2, x, domain='EX'), x)
(0, 1/(x**2 + y**2))
>>> ratint_ratpart(Poly(36, x, domain='ZZ'),
... Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x, domain='ZZ'), x)
((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2))
See Also
========
ratint, ratint_logpart
"""
from sympy.solvers.solvers import solve
f = Poly(f, x)
g = Poly(g, x)
u, v, _ = g.cofactors(g.diff())
n = u.degree()
m = v.degree()
A_coeffs = [Dummy('a' + str(n - i)) for i in range(0, n)]
B_coeffs = [Dummy('b' + str(m - i)) for i in range(0, m)]
C_coeffs = A_coeffs + B_coeffs
A = Poly(A_coeffs, x, domain=ZZ[C_coeffs])
B = Poly(B_coeffs, x, domain=ZZ[C_coeffs])
H = f - A.diff() * v + A * (u.diff() * v).quo(u) - B * u
result = solve(H.coeffs(), C_coeffs)
A = A.as_expr().subs(result)
B = B.as_expr().subs(result)
rat_part = cancel(A / u.as_expr(), x)
log_part = cancel(B / v.as_expr(), x)
return rat_part, log_part
def _solve_as_poly(f, symbol, domain=S.Complexes):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f,
symbol,
cubics=True,
quartics=True,
quintics=True,
domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
else:
poly = Poly(f)
if poly is None:
result = ConditionSet(symbol, Eq(f, 0), domain)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(
*roots(poly, cubics=True, quartics=True, quintics=True).keys())
if len(poly_solns) < deg:
result = ConditionSet(symbol, Eq(f, 0), domain)
if gen != symbol:
y = Dummy('y')
inverter = invert_real if domain.is_subset(
S.Reals) else invert_complex
lhs, rhs_s = inverter(gen, y, symbol)
if lhs == symbol:
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with free
# variables because that makes the solution more complicated. For
# example expand_complex(a) returns re(a) + I*im(a)
if all([
s.free_symbols == set() and not isinstance(s, RootOf)
for s in result
]):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
if isinstance(result, FiniteSet):
result = result.intersection(domain)
return result
else:
return ConditionSet(symbol, Eq(f, 0), domain)
def _solve_as_poly(f, symbol, solveset_solver, invert_func):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f,
symbol,
cubics=True,
quartics=True,
quintics=True,
domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
raise NotImplementedError("Couldn't find all roots "
"of the equation %s" % f)
else:
poly = Poly(f)
if poly is None:
raise NotImplementedError("Could not convert %s to Poly" % f)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(
*roots(poly, cubics=True, quartics=True, quintics=True).keys())
if len(poly_solns) < deg:
raise NotImplementedError("Couldn't find all the roots of "
"the equation %s" % f)
if gen != symbol:
y = Dummy('y')
lhs, rhs_s = invert_func(gen, y, symbol)
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
if lhs is not symbol and lhs is not gen:
result = Union(
*
[solveset_solver(lhs - rhs, symbol) for rhs in result])
else:
raise NotImplementedError
else:
raise NotImplementedError("multiple generators not handled"
" by solveset")
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with free
# variables because that makes the solution more complicated. For
# example expand_complex(a) returns re(a) + I*im(a)
if all([
s.free_symbols == set() and not isinstance(s, RootOf)
for s in result
]):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
return result
else:
raise NotImplementedError
