def _eval_nseries(self, x, n, logx, cdir=0):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy import im, cancel, I, Order, logcombine
if not logx:
logx = log(x)
if self.args[0] == x:
return logx
arg = self.args[0]
k, l = Wild("k"), Wild("l")
r = arg.match(k * x**l)
if r is not None:
k, l = r[k], r[l]
if l != 0 and not l.has(x) and not k.has(x):
r = log(k) + l * logx # XXX true regardless of assumptions?
return r
# TODO new and probably slow
try:
a, b = arg.leadterm(x)
s = arg.nseries(x, n=n + b, logx=logx)
except (ValueError, NotImplementedError):
s = arg.nseries(x, n=n, logx=logx)
while s.is_Order:
n += 1
s = arg.nseries(x, n=n, logx=logx)
a, b = s.removeO().leadterm(x)
p = cancel(s / (a * x**b) - 1)
if p.has(exp):
p = logcombine(p)
g = None
l = []
for i in range(n + 2):
g = log.taylor_term(i, p, g)
g = g.nseries(x, n=n, logx=logx)
l.append(g)
res = log(a) + b * logx
if cdir != 0:
cdir = self.args[0].dir(x, cdir)
if a.is_real and a.is_negative and im(cdir) < 0:
res -= 2 * I * S.Pi
return res + Add(*l) + Order(p**n, x)
def _eval_nseries(self, x, n, logx, cdir=0):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy import im, cancel, I, Order, logcombine
from itertools import product
if not logx:
logx = log(x)
if self.args[0] == x:
return logx
arg = self.args[0]
k, l = Wild("k"), Wild("l")
r = arg.match(k * x**l)
if r is not None:
k, l = r[k], r[l]
if l != 0 and not l.has(x) and not k.has(x):
r = log(k) + l * logx # XXX true regardless of assumptions?
return r
def coeff_exp(term, x):
coeff, exp = S.One, S.Zero
for factor in Mul.make_args(term):
if factor.has(x):
base, exp = factor.as_base_exp()
if base != x:
try:
return term.leadterm(x)
except ValueError:
return term, S.Zero
else:
coeff *= factor
return coeff, exp
# TODO new and probably slow
try:
a, b = arg.leadterm(x)
s = arg.nseries(x, n=n + b, logx=logx)
except (ValueError, NotImplementedError):
s = arg.nseries(x, n=n, logx=logx)
while s.is_Order:
n += 1
s = arg.nseries(x, n=n, logx=logx)
a, b = s.removeO().leadterm(x)
p = cancel(s / (a * x**b) - 1).expand().powsimp()
if p.has(exp):
p = logcombine(p)
if isinstance(p, Order):
n = p.getn()
_, d = coeff_exp(p, x)
if not d.is_positive:
return log(a) + b * logx + Order(x**n, x)
def mul(d1, d2):
res = {}
for e1, e2 in product(d1, d2):
ex = e1 + e2
if ex < n:
res[ex] = res.get(ex, S.Zero) + d1[e1] * d2[e2]
return res
pterms = {}
for term in Add.make_args(p):
co1, e1 = coeff_exp(term, x)
pterms[e1] = pterms.get(e1, S.Zero) + co1.removeO()
k = S.One
terms = {}
pk = pterms
while k * d < n:
coeff = -(-1)**k / k
for ex in pk:
terms[ex] = terms.get(ex, S.Zero) + coeff * pk[ex]
pk = mul(pk, pterms)
k += S.One
res = log(a) + b * logx
for ex in terms:
res += terms[ex] * x**(ex)
if cdir != 0:
cdir = self.args[0].dir(x, cdir)
if a.is_real and a.is_negative and im(cdir) < 0:
res -= 2 * I * S.Pi
return res + Order(x**n, x)