Source code for sympy.series.limits

from __future__ import print_function, division

from sympy.core import S, Symbol, Add, sympify, Expr, PoleError, Mul
from sympy.core.compatibility import string_types
from sympy.core.symbol import Dummy
from sympy.functions.combinatorial.factorials import factorial
from sympy.core.numbers import GoldenRatio
from sympy.functions.combinatorial.numbers import fibonacci
from sympy.functions.special.gamma_functions import gamma
from sympy.series.order import Order
from .gruntz import gruntz
from sympy.core.exprtools import factor_terms
from sympy.simplify.ratsimp import ratsimp
from sympy.polys import PolynomialError

[docs]def limit(e, z, z0, dir="+"): """ Compute the limit of e(z) at the point z0. z0 can be any expression, including oo and -oo. For dir="+" (default) it calculates the limit from the right (z->z0+) and for dir="-" the limit from the left (z->z0-). For infinite z0 (oo or -oo), the dir argument is determined from the direction of the infinity (i.e., dir="-" for oo). Examples ======== >>> from sympy import limit, sin, Symbol, oo >>> from import x >>> limit(sin(x)/x, x, 0) 1 >>> limit(1/x, x, 0, dir="+") oo >>> limit(1/x, x, 0, dir="-") -oo >>> limit(1/x, x, oo) 0 Notes ===== First we try some heuristics for easy and frequent cases like "x", "1/x", "x**2" and similar, so that it's fast. For all other cases, we use the Gruntz algorithm (see the gruntz() function). """ return Limit(e, z, z0, dir).doit(deep=False)
def heuristics(e, z, z0, dir): rv = None if abs(z0) is S.Infinity: rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-") if isinstance(rv, Limit): return elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function: r = [] for a in e.args: l = limit(a, z, z0, dir) if l.has(S.Infinity) and l.is_finite is None: return elif isinstance(l, Limit): return elif l is S.NaN: return else: r.append(l) if r: rv = e.func(*r) if rv is S.NaN: try: rat_e = ratsimp(e) except PolynomialError: return if rat_e is S.NaN or rat_e == e: return return limit(rat_e, z, z0, dir) return rv
[docs]class Limit(Expr): """Represents an unevaluated limit. Examples ======== >>> from sympy import Limit, sin, Symbol >>> from import x >>> Limit(sin(x)/x, x, 0) Limit(sin(x)/x, x, 0) >>> Limit(1/x, x, 0, dir="-") Limit(1/x, x, 0, dir='-') """ def __new__(cls, e, z, z0, dir="+"): e = sympify(e) z = sympify(z) z0 = sympify(z0) if z0 is S.Infinity: dir = "-" elif z0 is S.NegativeInfinity: dir = "+" if isinstance(dir, string_types): dir = Symbol(dir) elif not isinstance(dir, Symbol): raise TypeError("direction must be of type basestring or Symbol, not %s" % type(dir)) if str(dir) not in ('+', '-'): raise ValueError( "direction must be either '+' or '-', not %s" % dir) obj = Expr.__new__(cls) obj._args = (e, z, z0, dir) return obj @property def free_symbols(self): e = self.args[0] isyms = e.free_symbols isyms.difference_update(self.args[1].free_symbols) isyms.update(self.args[2].free_symbols) return isyms
[docs] def doit(self, **hints): """Evaluates limit""" from sympy.series.limitseq import limit_seq from sympy.functions import RisingFactorial e, z, z0, dir = self.args if hints.get('deep', True): e = e.doit(**hints) z = z.doit(**hints) z0 = z0.doit(**hints) if e == z: return z0 if not e.has(z): return e # gruntz fails on factorials but works with the gamma function # If no factorial term is present, e should remain unchanged. # factorial is defined to be zero for negative inputs (which # differs from gamma) so only rewrite for positive z0. if z0.is_positive: e = e.rewrite([factorial, RisingFactorial], gamma) if e.is_Mul: if abs(z0) is S.Infinity: e = factor_terms(e) e = e.rewrite(fibonacci, GoldenRatio) ok = lambda w: (z in w.free_symbols and any(a.is_polynomial(z) or any(z in m.free_symbols and m.is_polynomial(z) for m in Mul.make_args(a)) for a in Add.make_args(w))) if all(ok(w) for w in e.as_numer_denom()): u = Dummy(positive=True) if z0 is S.NegativeInfinity: inve = e.subs(z, -1/u) else: inve = e.subs(z, 1/u) r = limit(inve.as_leading_term(u), u, S.Zero, "+") if isinstance(r, Limit): return self else: return r if e.is_Order: return Order(limit(e.expr, z, z0), *e.args[1:]) try: r = gruntz(e, z, z0, dir) if r is S.NaN: raise PoleError() except (PoleError, ValueError): r = heuristics(e, z, z0, dir) if r is None: return self except NotImplementedError: # Trying finding limits of sequences if hints.get('sequence', True) and z0 is S.Infinity: trials = hints.get('trials', 5) r = limit_seq(e, z, trials) if r is None: raise NotImplementedError() else: raise NotImplementedError() return r