def _invert_real(f, g_ys, symbol):
""" Helper function for invert_real """
if not f.has(symbol):
raise ValueError("Inverse of constant function doesn't exist")
if f is symbol:
return (f, g_ys)
n = Dummy('n')
if hasattr(f, 'inverse') and not isinstance(f, TrigonometricFunction) and \
not isinstance(f, HyperbolicFunction):
if len(f.args) > 1:
raise ValueError("Only functions with one argument are supported.")
return _invert_real(f.args[0], imageset(Lambda(n,
f.inverse()(n)), g_ys),
symbol)
if isinstance(f, Abs):
return _invert_real(
f.args[0],
Union(
imageset(Lambda(n, n), g_ys).intersect(Interval(0, oo)),
imageset(Lambda(n, -n), g_ys).intersect(Interval(-oo, 0))),
symbol)
if f.is_Add:
# f = g + h
g, h = f.as_independent(symbol)
if g != S.Zero:
return _invert_real(h, imageset(Lambda(n, n - g), g_ys), symbol)
if f.is_Mul:
# f = g*h
g, h = f.as_independent(symbol)
if g != S.One:
return _invert_real(h, imageset(Lambda(n, n / g), g_ys), symbol)
if f.is_Pow:
base, expo = f.args
base_has_sym = base.has(symbol)
expo_has_sym = expo.has(symbol)
if not expo_has_sym:
res = imageset(Lambda(n, real_root(n, expo)), g_ys)
if expo.is_rational:
numer, denom = expo.as_numer_denom()
if numer == S.One or numer == -S.One:
return _invert_real(base, res, symbol)
else:
if numer % 2 == 0:
n = Dummy('n')
neg_res = imageset(Lambda(n, -n), res)
return _invert_real(base, res + neg_res, symbol)
else:
return _invert_real(base, res, symbol)
else:
if not base.is_positive:
raise ValueError("x**w where w is irrational is not "
"defined for negative x")
return _invert_real(base, res, symbol)
if not base_has_sym:
return _invert_real(expo,
imageset(Lambda(n,
log(n) / log(base)), g_ys),
symbol)
if isinstance(f, sin):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
sin_invs = Union(*[imageset(Lambda(n, n*pi + (-1)**n*asin(g_y)), \
S.Integers) for g_y in g_ys])
return _invert_real(f.args[0], sin_invs, symbol)
if isinstance(f, csc):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
csc_invs = Union(*[imageset(Lambda(n, n*pi + (-1)**n*acsc(g_y)), \
S.Integers) for g_y in g_ys])
return _invert_real(f.args[0], csc_invs, symbol)
if isinstance(f, cos):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
cos_invs_f1 = Union(*[imageset(Lambda(n, 2*n*pi + acos(g_y)), \
S.Integers) for g_y in g_ys])
cos_invs_f2 = Union(*[imageset(Lambda(n, 2*n*pi - acos(g_y)), \
S.Integers) for g_y in g_ys])
cos_invs = Union(cos_invs_f1, cos_invs_f2)
return _invert_real(f.args[0], cos_invs, symbol)
if isinstance(f, sec):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
sec_invs_f1 = Union(*[imageset(Lambda(n, 2*n*pi + asec(g_y)), \
S.Integers) for g_y in g_ys])
sec_invs_f2 = Union(*[imageset(Lambda(n, 2*n*pi - asec(g_y)), \
S.Integers) for g_y in g_ys])
sec_invs = Union(sec_invs_f1, sec_invs_f2)
return _invert_real(f.args[0], sec_invs, symbol)
if isinstance(f, tan) or isinstance(f, cot):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
tan_cot_invs = Union(*[imageset(Lambda(n, n*pi + f.inverse()(g_y)), \
S.Integers) for g_y in g_ys])
return _invert_real(f.args[0], tan_cot_invs, symbol)
return (f, g_ys)
def _invert_real(f, g_ys, symbol):
""" Helper function for invert_real """
if not f.has(symbol):
raise ValueError("Inverse of constant function doesn't exist")
if f is symbol:
return (f, g_ys)
n = Dummy('n')
if hasattr(f, 'inverse') and not isinstance(f, TrigonometricFunction) and \
not isinstance(f, HyperbolicFunction):
if len(f.args) > 1:
raise ValueError("Only functions with one argument are supported.")
return _invert_real(f.args[0],
imageset(Lambda(n, f.inverse()(n)), g_ys), symbol)
if isinstance(f, Abs):
return _invert_real(f.args[0],
Union(imageset(Lambda(n, n), g_ys).intersect(Interval(0, oo)),
imageset(Lambda(n, -n), g_ys).intersect(Interval(-oo, 0))),
symbol)
if f.is_Add:
# f = g + h
g, h = f.as_independent(symbol)
if g != S.Zero:
return _invert_real(h, imageset(Lambda(n, n - g), g_ys), symbol)
if f.is_Mul:
# f = g*h
g, h = f.as_independent(symbol)
if g != S.One:
return _invert_real(h, imageset(Lambda(n, n/g), g_ys), symbol)
if f.is_Pow:
base, expo = f.args
base_has_sym = base.has(symbol)
expo_has_sym = expo.has(symbol)
if not expo_has_sym:
res = imageset(Lambda(n, real_root(n, expo)), g_ys)
if expo.is_rational:
numer, denom = expo.as_numer_denom()
if numer == S.One or numer == - S.One:
return _invert_real(base, res, symbol)
else:
if numer % 2 == 0:
n = Dummy('n')
neg_res = imageset(Lambda(n, -n), res)
return _invert_real(base, res + neg_res, symbol)
else:
return _invert_real(base, res, symbol)
else:
if not base.is_positive:
raise ValueError("x**w where w is irrational is not "
"defined for negative x")
return _invert_real(base, res, symbol)
if not base_has_sym:
return _invert_real(expo, imageset(Lambda(n, log(n)/log(base)),
g_ys), symbol)
if isinstance(f, sin):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
sin_invs = Union(*[imageset(Lambda(n, n*pi + (-1)**n*asin(g_y)), \
S.Integers) for g_y in g_ys])
return _invert_real(f.args[0], sin_invs, symbol)
if isinstance(f, csc):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
csc_invs = Union(*[imageset(Lambda(n, n*pi + (-1)**n*acsc(g_y)), \
S.Integers) for g_y in g_ys])
return _invert_real(f.args[0], csc_invs, symbol)
if isinstance(f, cos):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
cos_invs_f1 = Union(*[imageset(Lambda(n, 2*n*pi + acos(g_y)), \
S.Integers) for g_y in g_ys])
cos_invs_f2 = Union(*[imageset(Lambda(n, 2*n*pi - acos(g_y)), \
S.Integers) for g_y in g_ys])
cos_invs = Union(cos_invs_f1, cos_invs_f2)
return _invert_real(f.args[0], cos_invs, symbol)
if isinstance(f, sec):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
sec_invs_f1 = Union(*[imageset(Lambda(n, 2*n*pi + asec(g_y)), \
S.Integers) for g_y in g_ys])
sec_invs_f2 = Union(*[imageset(Lambda(n, 2*n*pi - asec(g_y)), \
S.Integers) for g_y in g_ys])
sec_invs = Union(sec_invs_f1, sec_invs_f2)
return _invert_real(f.args[0], sec_invs, symbol)
if isinstance(f, tan) or isinstance(f, cot):
n = Dummy('n')
if isinstance(g_ys, FiniteSet):
tan_cot_invs = Union(*[imageset(Lambda(n, n*pi + f.inverse()(g_y)), \
S.Integers) for g_y in g_ys])
return _invert_real(f.args[0], tan_cot_invs, symbol)
return (f, g_ys)