Source code for sympy.series.limits

from sympy.core import S, Add, sympify, Expr, PoleError, Mul, oo, C
from gruntz import gruntz
from sympy.functions import sign, tan, cot

[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 doesn't matter. 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 Strategy: 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). """ from sympy import Wild, log e = sympify(e) z = sympify(z) z0 = sympify(z0) if e == z: return z0 if e.is_Rational: return e if not e.has(z): return e if e.func is tan: # discontinuity at odd multiples of pi/2; 0 at even disc = S.Pi/2 sign = 1 if dir == '-': sign *= -1 i = limit(sign*e.args[0], z, z0)/disc if i.is_integer: if i.is_even: return S.Zero elif i.is_odd: if dir == '+': return S.NegativeInfinity else: return S.Infinity if e.func is cot: # discontinuity at multiples of pi; 0 at odd pi/2 multiples disc = S.Pi sign = 1 if dir == '-': sign *= -1 i = limit(sign*e.args[0], z, z0)/disc if i.is_integer: if dir == '-': return S.NegativeInfinity else: return S.Infinity elif (2*i).is_integer: return S.Zero if e.is_Pow: b, ex = e.args c = None # records sign of b if b is +/-z or has a bounded value if b.is_Mul: c, b = b.as_two_terms() if c is S.NegativeOne and b == z: c = '-' elif b == z: c = '+' if ex.is_number: if c is None: base = b.subs(z, z0) if base.is_bounded and (ex.is_bounded or base is not S.One): return base**ex else: if z0 == 0 and ex < 0: if dir != c: # integer if ex.is_even: return S.Infinity elif ex.is_odd: return S.NegativeInfinity # rational elif ex.is_Rational: return (S.NegativeOne**ex)*S.Infinity else: return S.ComplexInfinity return S.Infinity return z0**ex if e.is_Mul or not z0 and e.is_Pow and b.func is log: if e.is_Mul: # weed out the z-independent terms i, d = e.as_independent(z) if i is not S.One and i.is_bounded: return i*limit(d, z, z0, dir) else: i, d = S.One, e if not z0: # look for log(z)**q or z**p*log(z)**q p, q = Wild("p"), Wild("q") r = d.match(z**p * log(z)**q) if r: p, q = [r.get(w, w) for w in [p, q]] if q and q.is_number and p.is_number: if q > 0: if p > 0: return S.Zero else: return -oo*i else: if p >= 0: return S.Zero else: return -oo*i if e.is_Add: if e.is_polynomial() and not z0.is_unbounded: return Add(*[limit(term, z, z0, dir) for term in e.args]) # this is a case like limit(x*y+x*z, z, 2) == x*y+2*x # but we need to make sure, that the general gruntz() algorithm is # executed for a case like "limit(sqrt(x+1)-sqrt(x),x,oo)==0" unbounded = []; unbounded_result=[] finite = []; unknown = [] ok = True for term in e.args: if not term.has(z) and not term.is_unbounded: finite.append(term) continue result = term.subs(z, z0) bounded = result.is_bounded if bounded is False or result is S.NaN: if unknown: ok = False break unbounded.append(term) if result != S.NaN: # take result from direction given result = limit(term, z, z0, dir) unbounded_result.append(result) elif bounded: finite.append(result) else: if unbounded: ok = False break unknown.append(result) if not ok: # we won't be able to resolve this with unbounded # terms, e.g. Sum(1/k, (k, 1, n)) - log(n) as n -> oo: # since the Sum is unevaluated it's boundedness is # unknown and the log(n) is oo so you get Sum - oo # which is unsatisfactory. raise NotImplementedError('unknown boundedness for %s' % (unknown or result)) u = Add(*unknown) if unbounded: inf_limit = Add(*unbounded_result) if inf_limit is not S.NaN: return inf_limit + u if finite: return Add(*finite) + limit(Add(*unbounded), z, z0, dir) + u else: return Add(*finite) + u if e.is_Order: args = e.args return C.Order(limit(args[0], z, z0), *args[1:]) try: r = gruntz(e, z, z0, dir) if r is S.NaN: raise PoleError() except PoleError: r = heuristics(e, z, z0, dir) return r
def heuristics(e, z, z0, dir): if z0 == oo: return limit(e.subs(z, 1/z), z, sympify(0), "+") elif e.is_Mul: r = [] for a in e.args: if not a.is_bounded: r.append(a.limit(z, z0, dir)) if r: return Mul(*r) elif e.is_Add: r = [] for a in e.args: r.append(a.limit(z, z0, dir)) return Add(*r) elif e.is_Function: return e.subs(e.args[0], limit(e.args[0], z, z0, dir)) msg = "Don't know how to calculate the limit(%s, %s, %s, dir=%s), sorry." raise PoleError(msg % (e, z, z0, dir)) 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) obj = Expr.__new__(cls) obj._args = (e, z, z0, dir) return obj def doit(self, **hints): e, z, z0, dir = self.args if hints.get('deep', True): e = e.doit(**hints) z = z.doit(**hints) z0 = z0.doit(**hints) return limit(e, z, z0, dir)