def limitinf(e, x):
"""Limit e(x) for x-> oo"""
if not e.has(x): return e #e is a constant
#this is needed to simplify 1/log(exp(-x**2))*log(exp(x**2)) to 1
if e.has(log):
e = e.normal()
c0, e0 = mrv_leadterm(e,x)
sig=sign(e0,x)
if sig==1: return S.Zero # e0>0: lim f = 0
elif sig==-1: #e0<0: lim f = +-oo (the sign depends on the sign of c0)
if c0.match(I*C.Wild("a", exclude=[I])):
return c0*oo
s = sign(c0, x)
#the leading term shouldn't be 0:
assert s != 0
return s * oo
elif sig==0: return limitinf(c0, x) #e0=0: lim f = lim c0
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR: return L * sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL: return R * sL
if f.is_Add:
L, R = getab(f)
lrsum = telescopic(L, R, (i, a, b))
if lrsum: return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
return lsum + rsum
# Polynomial terms with Faulhaber's formula
p = C.Wild('p')
e = f.match(i**p)
if e != None:
c = p.subs(e)
B = C.bernoulli
if c.is_integer and c >= 0:
s = (B(c + 1, b + 1) - B(c + 1, a)) / (c + 1)
return s.expand()
# Geometric terms
c1 = C.Wild('c1', exclude=[i])
c2 = C.Wild('c2', exclude=[i])
c3 = C.Wild('c3', exclude=[i])
e = f.match(c1**(c2 * i + c3))
if e is not None:
c1 = c1.subs(e)
c2 = c2.subs(e)
def eval_sum_symbolic(f, limits):
(i, a, b) = limits
if not f.has(i):
return f * (b - a + 1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR: return L * sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL: return R * sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
return lsum + rsum
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
return ((C.bernoulli(n + 1, b + 1) - C.bernoulli(n + 1, a)) /
(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return C.harmonic(b) - C.harmonic(a - 1)
else:
return C.harmonic(b, abs(n)) - C.harmonic(a - 1, abs(n))
# Geometric terms
c1 = C.Wild('c1', exclude=[i])
c2 = C.Wild('c2', exclude=[i])
c3 = C.Wild('c3', exclude=[i])
e = f.match(c1**(c2 * i + c3))
if e is not None:
c1 = c1.subs(e)
c2 = c2.subs(e)
c3 = c3.subs(e)
# TODO: more general limit handling
return c1**c3 * (c1**(a * c2) - c1**(c2 + b * c2)) / (1 - c1**c2)
r = gosper_sum(f, (i, a, b))
if not r in (None, S.NaN):
return r
return eval_sum_hyper(f, (i, a, b))
def eval_sum_symbolic(f, limits):
(i, a, b) = limits
if not f.has(i):
return f * (b - a + 1)
# Linearity
if f.is_Mul:
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L * sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return R * sL
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
return lsum + rsum
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
return ((C.bernoulli(n + 1, b + 1) - C.bernoulli(n + 1, a)) /
(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return C.harmonic(b) - C.harmonic(a - 1)
else:
return C.harmonic(b, abs(n)) - C.harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity)
or b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = C.Wild('c1', exclude=[i])
c2 = C.Wild('c2', exclude=[i])
c3 = C.Wild('c3', exclude=[i])
e = f.match(c1**(c2 * i + c3))
if e is not None:
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p * (q**a - q**(b + 1)) / (1 - q)
l = p * (b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if not r in (None, S.NaN):
return r
return eval_sum_hyper(f, (i, a, b))
