def _eval_product(self, term=None):
k = self.index
a = self.lower
n = self.upper
if term is None:
term = self.term
if not term.has(k):
return term**(n - a + 1)
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
C_ = poly.LC
all_roots = roots(poly, multiple=True)
for r in all_roots:
A *= C.RisingFactorial(a - r, n - a + 1)
Q *= n - r
if len(all_roots) < poly.degree:
B = Product(quo(poly, Q.as_poly(k)), (k, a, n))
return poly.LC**(n - a + 1) * A * B
elif term.is_Add:
p, q = term.as_numer_denom()
p = self._eval_product(p)
q = self._eval_product(q)
return p / q
elif term.is_Mul:
exclude, include = [], []
for t in term.args:
p = self._eval_product(t)
if p is not None:
exclude.append(p)
else:
include.append(p)
if not exclude:
return None
else:
A, B = Mul(*exclude), Mul(*include)
return A * Product(B, (k, a, n))
elif term.is_Pow:
if not term.base.has(k):
s = sum(term.exp, (k, a, n))
if not isinstance(s, Sum):
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(term.base)
if p is not None:
return p**term.exp
def _eval_product(self, a, n, term):
from sympy import summation, Sum
k = self.index
if not term.has(k):
return term**(n - a + 1)
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
C_ = poly.LC()
all_roots = roots(poly, multiple=True)
for r in all_roots:
A *= C.RisingFactorial(a - r, n - a + 1)
Q *= n - r
if len(all_roots) < poly.degree():
B = Product(quo(poly, Q.as_poly(k)), (k, a, n))
return poly.LC()**(n - a + 1) * A * B
elif term.is_Add:
p, q = term.as_numer_denom()
p = self._eval_product(a, n, p)
q = self._eval_product(a, n, q)
return p / q
elif term.is_Mul:
exclude, include = [], []
for t in term.args:
p = self._eval_product(a, n, t)
if p is not None:
exclude.append(p)
else:
include.append(t)
if not exclude:
return None
else:
A, B = Mul(*exclude), term._new_rawargs(*include)
return A * Product(B, (k, a, n))
elif term.is_Pow:
if not term.base.has(k):
s = summation(term.exp, (k, a, n))
if not isinstance(s, Sum):
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(a, n, term.base)
if p is not None:
return p**term.exp
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return -p * (n-2)**2/(n*(n-1)) * x**2
else:
k = (n - 1) // 2
R = C.RisingFactorial(S.Half, k)
F = C.factorial(k)
return (-1)**k * R / F * x**n / n
def _eval_expand_func(self, deep=True, **hints):
if deep:
arg = self.args[0].expand(deep, **hints)
else:
arg = self.args[0]
if arg.is_Add:
coeff, tail = arg.as_coeff_add()
if coeff and coeff.q != 1:
tail = (C.Rational(1, coeff.q), ) + tail
coeff = floor(coeff)
tail = arg._new_rawargs(*tail, **dict(reeval=False))
return gamma(tail) * C.RisingFactorial(tail, coeff)
return self.func(*self.args)
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi*S.ImaginaryUnit / 2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p * (n-2)**2/(n*(n-1)) * x**2
else:
k = (n - 1) // 2
R = C.RisingFactorial(S.Half, k)
F = C.Factorial(k)
return -R / F * S.ImaginaryUnit * x**n / n
def _eval_expand_func(self, *args):
arg = self.args[0].expand()
if arg.is_Add:
for i, coeff in enumerate(arg.args[:]):
if arg.args[i].is_Number:
terms = C.Add(*(arg.args[:i] + arg.args[i + 1:]))
if coeff.is_Rational:
if coeff.q != 1:
terms += C.Rational(1, coeff.q)
coeff = C.Integer(int(coeff))
else:
continue
return gamma(terms) * C.RisingFactorial(terms, coeff)
return self.func(*self.args)
def _eval_expand_func(self, **hints):
arg = self.args[0]
if arg.is_Rational:
if abs(arg.p) > arg.q:
x = Dummy('x')
n = arg.p // arg.q
p = arg.p - n*arg.q
return gamma(x + n)._eval_expand_func().subs(x, Rational(p, arg.q))
if arg.is_Add:
coeff, tail = arg.as_coeff_add()
if coeff and coeff.q != 1:
intpart = floor(coeff)
tail = (coeff - intpart,) + tail
coeff = intpart
tail = arg._new_rawargs(*tail, reeval=False)
return gamma(tail)*C.RisingFactorial(tail, coeff)
return self.func(*self.args)
def _eval_product(self, term, limits):
from sympy import summation
(k, a, n) = limits
if k not in term.free_symbols:
return term**(n - a + 1)
if a == n:
return term.subs(k, a)
dif = n - a
if dif.is_Integer:
return Mul(*[term.subs(k, a + i) for i in xrange(dif + 1)])
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
all_roots = roots(poly, multiple=True)
for r in all_roots:
A *= C.RisingFactorial(a - r, n - a + 1)
Q *= n - r
if len(all_roots) < poly.degree():
arg = quo(poly, Q.as_poly(k))
B = Product(arg, (k, a, n)).doit()
return poly.LC()**(n - a + 1) * A * B
elif term.is_Add:
p, q = term.as_numer_denom()
p = self._eval_product(p, (k, a, n))
q = self._eval_product(q, (k, a, n))
return p / q
elif term.is_Mul:
exclude, include = [], []
for t in term.args:
p = self._eval_product(t, (k, a, n))
if p is not None:
exclude.append(p)
else:
include.append(t)
if not exclude:
return None
else:
arg = term._new_rawargs(*include)
A = Mul(*exclude)
B = Product(arg, (k, a, n)).doit()
return A * B
elif term.is_Pow:
if not term.base.has(k):
s = summation(term.exp, (k, a, n))
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(term.base, (k, a, n))
if p is not None:
return p**term.exp
elif isinstance(term, Product):
evaluated = term.doit()
f = self._eval_product(evaluated, limits)
if f is None:
return Product(evaluated, limits)
else:
return f
