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366 lines
12 KiB
366 lines
12 KiB
from sympy.calculus.accumulationbounds import AccumBounds
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from sympy.core import S, Symbol, Add, sympify, Expr, PoleError, Mul
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from sympy.core.exprtools import factor_terms
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from sympy.core.numbers import Float
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from sympy.functions.combinatorial.factorials import factorial
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from sympy.functions.elementary.complexes import (Abs, sign)
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from sympy.functions.elementary.exponential import (exp, log)
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from sympy.functions.special.gamma_functions import gamma
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from sympy.polys import PolynomialError, factor
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from sympy.series.order import Order
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from sympy.simplify.powsimp import powsimp
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from sympy.simplify.ratsimp import ratsimp
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from sympy.simplify.simplify import nsimplify, together
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from .gruntz import gruntz
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def limit(e, z, z0, dir="+"):
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"""Computes the limit of ``e(z)`` at the point ``z0``.
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Parameters
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==========
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e : expression, the limit of which is to be taken
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z : symbol representing the variable in the limit.
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Other symbols are treated as constants. Multivariate limits
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are not supported.
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z0 : the value toward which ``z`` tends. Can be any expression,
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including ``oo`` and ``-oo``.
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dir : string, optional (default: "+")
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The limit is bi-directional if ``dir="+-"``, from the right
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(z->z0+) if ``dir="+"``, and from the left (z->z0-) if
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``dir="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``dir``
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argument is determined from the direction of the infinity
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(i.e., ``dir="-"`` for ``oo``).
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Examples
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========
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>>> from sympy import limit, sin, oo
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>>> from sympy.abc import x
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>>> limit(sin(x)/x, x, 0)
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1
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>>> limit(1/x, x, 0) # default dir='+'
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oo
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>>> limit(1/x, x, 0, dir="-")
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-oo
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>>> limit(1/x, x, 0, dir='+-')
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zoo
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>>> limit(1/x, x, oo)
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0
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Notes
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=====
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First we try some heuristics for easy and frequent cases like "x", "1/x",
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"x**2" and similar, so that it's fast. For all other cases, we use the
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Gruntz algorithm (see the gruntz() function).
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See Also
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========
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limit_seq : returns the limit of a sequence.
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"""
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return Limit(e, z, z0, dir).doit(deep=False)
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def heuristics(e, z, z0, dir):
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"""Computes the limit of an expression term-wise.
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Parameters are the same as for the ``limit`` function.
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Works with the arguments of expression ``e`` one by one, computing
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the limit of each and then combining the results. This approach
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works only for simple limits, but it is fast.
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"""
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rv = None
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if abs(z0) is S.Infinity:
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rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
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if isinstance(rv, Limit):
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return
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elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
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r = []
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for a in e.args:
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l = limit(a, z, z0, dir)
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if l.has(S.Infinity) and l.is_finite is None:
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if isinstance(e, Add):
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m = factor_terms(e)
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if not isinstance(m, Mul): # try together
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m = together(m)
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if not isinstance(m, Mul): # try factor if the previous methods failed
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m = factor(e)
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if isinstance(m, Mul):
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return heuristics(m, z, z0, dir)
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return
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return
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elif isinstance(l, Limit):
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return
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elif l is S.NaN:
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return
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else:
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r.append(l)
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if r:
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rv = e.func(*r)
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if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
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r2 = []
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e2 = []
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for ii in range(len(r)):
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if isinstance(r[ii], AccumBounds):
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r2.append(r[ii])
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else:
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e2.append(e.args[ii])
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if len(e2) > 0:
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e3 = Mul(*e2).simplify()
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l = limit(e3, z, z0, dir)
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rv = l * Mul(*r2)
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if rv is S.NaN:
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try:
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rat_e = ratsimp(e)
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except PolynomialError:
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return
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if rat_e is S.NaN or rat_e == e:
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return
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return limit(rat_e, z, z0, dir)
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return rv
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class Limit(Expr):
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"""Represents an unevaluated limit.
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Examples
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========
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>>> from sympy import Limit, sin
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>>> from sympy.abc import x
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>>> Limit(sin(x)/x, x, 0)
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Limit(sin(x)/x, x, 0)
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>>> Limit(1/x, x, 0, dir="-")
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Limit(1/x, x, 0, dir='-')
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"""
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def __new__(cls, e, z, z0, dir="+"):
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e = sympify(e)
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z = sympify(z)
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z0 = sympify(z0)
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if z0 is S.Infinity:
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dir = "-"
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elif z0 is S.NegativeInfinity:
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dir = "+"
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if(z0.has(z)):
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raise NotImplementedError("Limits approaching a variable point are"
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" not supported (%s -> %s)" % (z, z0))
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if isinstance(dir, str):
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dir = Symbol(dir)
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elif not isinstance(dir, Symbol):
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raise TypeError("direction must be of type basestring or "
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"Symbol, not %s" % type(dir))
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if str(dir) not in ('+', '-', '+-'):
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raise ValueError("direction must be one of '+', '-' "
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"or '+-', not %s" % dir)
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obj = Expr.__new__(cls)
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obj._args = (e, z, z0, dir)
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return obj
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@property
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def free_symbols(self):
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e = self.args[0]
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isyms = e.free_symbols
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isyms.difference_update(self.args[1].free_symbols)
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isyms.update(self.args[2].free_symbols)
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return isyms
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def pow_heuristics(self, e):
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_, z, z0, _ = self.args
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b1, e1 = e.base, e.exp
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if not b1.has(z):
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res = limit(e1*log(b1), z, z0)
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return exp(res)
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ex_lim = limit(e1, z, z0)
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base_lim = limit(b1, z, z0)
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if base_lim is S.One:
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if ex_lim in (S.Infinity, S.NegativeInfinity):
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res = limit(e1*(b1 - 1), z, z0)
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return exp(res)
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if base_lim is S.NegativeInfinity and ex_lim is S.Infinity:
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return S.ComplexInfinity
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def doit(self, **hints):
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"""Evaluates the limit.
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Parameters
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==========
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deep : bool, optional (default: True)
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Invoke the ``doit`` method of the expressions involved before
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taking the limit.
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hints : optional keyword arguments
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To be passed to ``doit`` methods; only used if deep is True.
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"""
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e, z, z0, dir = self.args
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if z0 is S.ComplexInfinity:
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raise NotImplementedError("Limits at complex "
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"infinity are not implemented")
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if hints.get('deep', True):
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e = e.doit(**hints)
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z = z.doit(**hints)
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z0 = z0.doit(**hints)
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if e == z:
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return z0
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if not e.has(z):
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return e
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if z0 is S.NaN:
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return S.NaN
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if e.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity, S.NaN):
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return self
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if e.is_Order:
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return Order(limit(e.expr, z, z0), *e.args[1:])
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cdir = 0
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if str(dir) == "+":
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cdir = 1
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elif str(dir) == "-":
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cdir = -1
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def set_signs(expr):
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if not expr.args:
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return expr
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newargs = tuple(set_signs(arg) for arg in expr.args)
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if newargs != expr.args:
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expr = expr.func(*newargs)
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abs_flag = isinstance(expr, Abs)
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sign_flag = isinstance(expr, sign)
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if abs_flag or sign_flag:
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sig = limit(expr.args[0], z, z0, dir)
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if sig.is_zero:
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sig = limit(1/expr.args[0], z, z0, dir)
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if sig.is_extended_real:
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if (sig < 0) == True:
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return -expr.args[0] if abs_flag else S.NegativeOne
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elif (sig > 0) == True:
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return expr.args[0] if abs_flag else S.One
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return expr
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if e.has(Float):
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# Convert floats like 0.5 to exact SymPy numbers like S.Half, to
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# prevent rounding errors which can lead to unexpected execution
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# of conditional blocks that work on comparisons
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# Also see comments in https://github.com/sympy/sympy/issues/19453
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e = nsimplify(e)
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e = set_signs(e)
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if e.is_meromorphic(z, z0):
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if abs(z0) is S.Infinity:
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newe = e.subs(z, 1/z)
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# cdir changes sign as oo- should become 0+
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cdir = -cdir
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else:
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newe = e.subs(z, z + z0)
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try:
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coeff, ex = newe.leadterm(z, cdir=cdir)
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except ValueError:
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pass
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else:
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if ex > 0:
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return S.Zero
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elif ex == 0:
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return coeff
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if cdir == 1 or not(int(ex) & 1):
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return S.Infinity*sign(coeff)
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elif cdir == -1:
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return S.NegativeInfinity*sign(coeff)
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else:
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return S.ComplexInfinity
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if abs(z0) is S.Infinity:
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if e.is_Mul:
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e = factor_terms(e)
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newe = e.subs(z, 1/z)
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# cdir changes sign as oo- should become 0+
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cdir = -cdir
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else:
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newe = e.subs(z, z + z0)
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try:
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coeff, ex = newe.leadterm(z, cdir=cdir)
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except (ValueError, NotImplementedError, PoleError):
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# The NotImplementedError catching is for custom functions
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e = powsimp(e)
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if e.is_Pow:
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r = self.pow_heuristics(e)
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if r is not None:
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return r
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else:
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if coeff.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity):
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return self
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if not coeff.has(z):
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if ex.is_positive:
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return S.Zero
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elif ex == 0:
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return coeff
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elif ex.is_negative:
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if ex.is_integer:
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if cdir == 1 or ex.is_even:
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return S.Infinity*sign(coeff)
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elif cdir == -1:
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return S.NegativeInfinity*sign(coeff)
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else:
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return S.ComplexInfinity
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else:
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if cdir == 1:
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return S.Infinity*sign(coeff)
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elif cdir == -1:
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return S.Infinity*sign(coeff)*S.NegativeOne**ex
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else:
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return S.ComplexInfinity
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# gruntz fails on factorials but works with the gamma function
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# If no factorial term is present, e should remain unchanged.
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# factorial is defined to be zero for negative inputs (which
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# differs from gamma) so only rewrite for positive z0.
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if z0.is_extended_positive:
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e = e.rewrite(factorial, gamma)
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l = None
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try:
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if str(dir) == '+-':
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r = gruntz(e, z, z0, '+')
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l = gruntz(e, z, z0, '-')
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if l != r:
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raise ValueError("The limit does not exist since "
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"left hand limit = %s and right hand limit = %s"
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% (l, r))
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else:
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r = gruntz(e, z, z0, dir)
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if r is S.NaN or l is S.NaN:
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raise PoleError()
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except (PoleError, ValueError):
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if l is not None:
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raise
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r = heuristics(e, z, z0, dir)
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if r is None:
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return self
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return r
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