def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if C.Order(1, x).contains(arg):
return 1 / arg
else:
return self.func(arg)
def extract_leading_order(self, symbols, point=None):
"""
Returns the leading term and its order.
Examples
========
>>> from sympy.abc import x
>>> (x + 1 + 1/x**5).extract_leading_order(x)
((x**(-5), O(x**(-5))),)
>>> (1 + x).extract_leading_order(x)
((1, O(1)),)
>>> (x + x**2).extract_leading_order(x)
((x, O(x)),)
"""
lst = []
symbols = list(symbols if is_sequence(symbols) else [symbols])
if not point:
point = [0]*len(symbols)
seq = [(f, C.Order(f, *zip(symbols, point))) for f in self.args]
for ef, of in seq:
for e, o in lst:
if o.contains(of) and o != of:
of = None
break
if of is None:
continue
new_lst = [(ef, of)]
for e, o in lst:
if of.contains(o) and o != of:
continue
new_lst.append((e, o))
lst = new_lst
return tuple(lst)
def _eval_as_leading_term(self, x):
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and C.Order(1, x).contains(arg):
return 1 / arg
else:
return self.func(arg)
def _eval_as_leading_term(self, x):
arg = self.args[0]
if arg.is_Add:
return C.Mul(*[exp(f).as_leading_term(x) for f in arg.args])
arg = self.args[0].as_leading_term(x)
if C.Order(1,x).contains(arg):
return S.One
return exp(arg)
def _taylor(self, x, x0, n):
l = []
g = None
for i in xrange(n):
g = self.taylor_term(i, self.args[0], g)
g = g.nseries(x, x0, n)
l.append(g)
return C.Add(*l) + C.Order(x**n, x)
def _eval_nseries(self, x, x0, 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, l = r[k], r[l]
if l != 0 and not l.has(x) and not k.has(x):
r = log(k) + l*log(x)
return r
order = C.Order(x**n, x)
arg = self.args[0]
x = order.symbols[0]
ln = C.log
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 ln(a).series(), but one needs to
# make sure that ln(sin(x)/x) doesn't get "simplified" to
# -log(x)+ln(sin(x)) and an infinite recursion occurs, see also the
# issue 252.
obj = ln(lt) + ln(a)._eval_nseries(x, x0, n)
else:
# arg -> arg0 + (arg - arg0) -> arg0 * (1 + (arg/arg0 - 1))
z = (arg/arg0 - 1)
x = order.symbols[0]
ln = C.log
o = C.Order(z, x)
if o is S.Zero:
return ln(1+z)+ ln(arg0)
if o.expr.is_number:
e = ln(order.expr*x)/ln(x)
else:
e = ln(order.expr)/ln(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 ln(1+z)+ ln(arg0)
try:
n = int(n)
except TypeError:
#well, the n is something more complicated (like 1+log(2))
n = int(n.evalf()) + 1
assert n>=0,`n`
l = []
g = None
for i in xrange(n+2):
g = ln.taylor_term(i, z, g)
g = g.nseries(x, x0, n)
l.append(g)
obj = C.Add(*l) + ln(arg0)
obj2 = expand_log(powsimp(obj, deep=True, combine='exp'))
if obj2 != obj:
r = obj2.nseries(x, x0, n)
else:
r = obj
if r == self:
return self
return r + order
