def _eval_nseries(self, x, n, logx):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy import cancel
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 = r.get(r, S.One)
#l = r.get(l, S.Zero)
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
s = self.args[0].nseries(x, n=n, logx=logx)
while s.is_Order:
n += 1
s = self.args[0].nseries(x, n=n, logx=logx)
a, b = s.leadterm(x)
p = cancel(s/(a*x**b) - 1)
g = None
l = []
for i in xrange(n + 2):
g = log.taylor_term(i, p, g)
g = g.nseries(x, n=n, logx=logx)
l.append(g)
return log(a) + b*logx + Add(*l) + C.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
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)
def _eval_nseries(self, x, n):
from sympy import powsimp
arg = self.args[0]
k, l = Wild("k"), Wild("l")
r = arg.match(k*x**l)
if r is not None:
#k = r.get(r, S.One)
#l = r.get(l, S.Zero)
k, l = r[k], r[l]
if l != 0 and not l.has(x) and not k.has(x):
r = log(k) + l*log(x) # XXX true regardless of assumptions?
return r
order = C.Order(x**n, x)
arg = self.args[0]
use_lt = not C.Order(1, x).contains(arg)
if not use_lt:
arg0 = arg.limit(x, 0)
use_lt = (arg0 is S.Zero)
if use_lt: # singularity, #example: self = log(sin(x))
# arg = (arg / lt) * lt
lt = arg.as_leading_term(x) # arg = sin(x); lt = x
a = powsimp((arg/lt).expand(), deep=True, combine='exp') # a = sin(x)/x
# the idea is to recursively call log(a).series(), but one needs to
# make sure that log(sin(x)/x) doesn't get "simplified" to
# -log(x)+log(sin(x)) and an infinite recursion occurs, see also the
# issue 252.
obj = log(lt) + log(a).nseries(x, n=n)
else:
# arg -> arg0 + (arg - arg0) -> arg0 * (1 + (arg/arg0 - 1))
z = (arg/arg0 - 1)
o = C.Order(z, x)
if o is S.Zero:
return log(1 + z) + log(arg0)
if o.expr.is_number:
e = log(order.expr*x)/log(x)
else:
e = log(order.expr)/log(o.expr)
n = e.limit(x, 0) + 1
if n.is_unbounded:
# requested accuracy gives infinite series,
# order is probably nonpolynomial e.g. O(exp(-1/x), x).
return log(1 + z) + log(arg0)
# XXX was int or floor intended? int used to behave like floor
try:
n = int(n)
except TypeError:
#well, the n is something more complicated (like 1+log(2))
n = int(n.evalf()) + 1 # XXX why is 1 being added?
assert n>=0, `n`
l = []
g = None
for i in xrange(n + 2):
g = log.taylor_term(i, z, g)
g = g.nseries(x, n=n)
l.append(g)
obj = Add(*l) + log(arg0)
obj2 = expand_log(powsimp(obj, deep=True, combine='exp'))
if obj2 != obj:
r = obj2.nseries(x, n=n)
else:
r = obj
if r == self:
return self
return r + order
def _eval_nseries(self, x, n):
from sympy import powsimp
arg = self.args[0]
k, l = Wild("k"), Wild("l")
r = arg.match(k * x**l)
if r is not None:
#k = r.get(r, S.One)
#l = r.get(l, S.Zero)
k, l = r[k], r[l]
if l != 0 and not l.has(x) and not k.has(x):
r = log(k) + l * log(x) # XXX true regardless of assumptions?
return r
order = C.Order(x**n, x)
arg = self.args[0]
use_lt = not C.Order(1, x).contains(arg)
if not use_lt:
arg0 = arg.limit(x, 0)
use_lt = (arg0 is S.Zero)
if use_lt: # singularity, #example: self = log(sin(x))
# arg = (arg / lt) * lt
lt = arg.as_leading_term(x) # arg = sin(x); lt = x
a = powsimp((arg / lt).expand(), deep=True,
combine='exp') # a = sin(x)/x
# the idea is to recursively call log(a).series(), but one needs to
# make sure that log(sin(x)/x) doesn't get "simplified" to
# -log(x)+log(sin(x)) and an infinite recursion occurs, see also the
# issue 252.
obj = log(lt) + log(a).nseries(x, n=n)
else:
# arg -> arg0 + (arg - arg0) -> arg0 * (1 + (arg/arg0 - 1))
z = (arg / arg0 - 1)
o = C.Order(z, x)
if o is S.Zero:
return log(1 + z) + log(arg0)
if o.expr.is_number:
e = log(order.expr * x) / log(x)
else:
e = log(order.expr) / log(o.expr)
n = e.limit(x, 0) + 1
if n.is_unbounded:
# requested accuracy gives infinite series,
# order is probably nonpolynomial e.g. O(exp(-1/x), x).
return log(1 + z) + log(arg0)
# XXX was int or floor intended? int used to behave like floor
try:
n = int(n)
except TypeError:
#well, the n is something more complicated (like 1+log(2))
n = int(n.evalf()) + 1 # XXX why is 1 being added?
assert n >= 0, ` n `
l = []
g = None
for i in xrange(n + 2):
g = log.taylor_term(i, z, g)
g = g.nseries(x, n=n)
l.append(g)
obj = Add(*l) + log(arg0)
obj2 = expand_log(powsimp(obj, deep=True, combine='exp'))
if obj2 != obj:
r = obj2.nseries(x, n=n)
else:
r = obj
if r == self:
return self
return r + order
