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3267: limit watches for difficult Add terms

In cases as noted in the code, try see if the rational
form of the terms gives a result, and if so continue with the limit algorithm.
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commit 626b5c9d7ab9dfbaa74210dc15b14e00531d489c 1 parent 5c33969
Christopher Smith smichr authored
7 sympy/integrals/tests/test_integrals.py
@@ -2,7 +2,7 @@
2 2 Function, Rational, log, sin, cos, pi, E, I, Poly, LambertW, diff, Matrix,
3 3 sympify, sqrt, atan, asin, acos, asinh, acosh, DiracDelta, Heaviside,
4 4 Lambda, sstr, Add, Tuple, Interval, Sum, factor, trigsimp, simplify, O,
5   - terms_gcd)
  5 + terms_gcd, EulerGamma, Ci)
6 6 from sympy.utilities.pytest import XFAIL, raises
7 7 from sympy.physics.units import m, s
8 8
@@ -684,10 +684,9 @@ def test_atom_bug():
684 684 assert heurisch(meijerg([], [], [1], [], x), x) is None
685 685
686 686 def test_limit_bug():
687   - # NOTE this used to raise NotImplementedError because of a limit problem.
688   - # actually gruntz() can do this limit, see issue 2079
689 687 assert integrate(sin(x*y*z), (x, 0, pi), (y, 0, pi)) == \
690   - Integral(-cos(pi*y*z)/(y*z) + 1/(y*z), (y, 0, pi))
  688 + -((-log(pi*z) + log(pi**2*z**2)/2 + Ci(pi**2*z)
  689 + )/z) + log(z**2)/(2*z) + EulerGamma/z + 2*log(pi)/z
691 690
692 691 # The following tests work using meijerint.
693 692 def test_issue841():
112 sympy/series/limits.py
@@ -161,42 +161,92 @@ def limit(e, z, z0, dir="+"):
161 161 # this is a case like limit(x*y+x*z, z, 2) == x*y+2*x
162 162 # but we need to make sure, that the general gruntz() algorithm is
163 163 # executed for a case like "limit(sqrt(x+1)-sqrt(x),x,oo)==0"
164   - unbounded = []; unbounded_result=[]
165   - finite = []; unknown = []
166   - ok = True
167   - for term in e.args:
168   - if not term.has(z) and not term.is_unbounded:
169   - finite.append(term)
170   - continue
171   - result = term.subs(z, z0)
172   - bounded = result.is_bounded
173   - if bounded is False or result is S.NaN:
174   - if unknown:
175   - ok = False
176   - break
177   - unbounded.append(term)
178   - if result != S.NaN:
179   - # take result from direction given
180   - result = limit(term, z, z0, dir)
181   - unbounded_result.append(result)
182   - elif bounded:
183   - finite.append(result)
  164 +
  165 + unbounded = []
  166 + unbounded_result = []
  167 + unbounded_const = []
  168 + unknown = []
  169 + unknown_result = []
  170 + finite = []
  171 + zero = []
  172 + def _sift(term):
  173 + if z not in term.free_symbols:
  174 + if term.is_unbounded:
  175 + unbounded_const.append(term)
  176 + else:
  177 + finite.append(term)
184 178 else:
185   - if unbounded:
186   - ok = False
187   - break
188   - unknown.append(result)
189   - if not ok:
  179 + result = term.subs(z, z0)
  180 + bounded = result.is_bounded
  181 + if bounded is False or result is S.NaN:
  182 + unbounded.append(term)
  183 + if result != S.NaN:
  184 + # take result from direction given
  185 + result = limit(term, z, z0, dir)
  186 + unbounded_result.append(result)
  187 + elif bounded:
  188 + if result:
  189 + finite.append(result)
  190 + else:
  191 + zero.append(term)
  192 + else:
  193 + unknown.append(term)
  194 + unknown_result.append(result)
  195 +
  196 + for term in e.args:
  197 + _sift(term)
  198 +
  199 + bad = bool(unknown and unbounded)
  200 + if bad or len(unknown) > 1 or len(unbounded) > 1 and not zero:
  201 + uu = unknown + unbounded
190 202 # we won't be able to resolve this with unbounded
191 203 # terms, e.g. Sum(1/k, (k, 1, n)) - log(n) as n -> oo:
192 204 # since the Sum is unevaluated it's boundedness is
193 205 # unknown and the log(n) is oo so you get Sum - oo
194   - # which is unsatisfactory.
195   - raise NotImplementedError('unknown boundedness for %s' %
196   - (unknown or result))
197   - u = Add(*unknown)
198   - if unbounded:
199   - inf_limit = Add(*unbounded_result)
  206 + # which is unsatisfactory. BUT...if there are both
  207 + # unknown and unbounded terms (condition 'bad') or
  208 + # there are multiple terms that are unknown, or
  209 + # there are multiple symbolic unbounded terms they may
  210 + # respond better if they are made into a rational
  211 + # function, so give them a chance to do so before
  212 + # reporting failure.
  213 + u = Add(*uu)
  214 + f = u.normal()
  215 + if f != u:
  216 + unknown = []
  217 + unbounded = []
  218 + unbounded_result = []
  219 + unknown_result = []
  220 + _sift(limit(f, z, z0, dir))
  221 +
  222 + # We came in with a) unknown and unbounded terms or b) had multiple
  223 + # unknown terms
  224 +
  225 + # At this point we've done one of 3 things.
  226 + # (1) We did nothing with f so we now report the error
  227 + # showing the troublesome terms which are now in uu. OR
  228 +
  229 + # (2) We did something with f but the result came back as unknown.
  230 + # Normally this wouldn't be a problem,
  231 + # but we had either multiple terms that were troublesome (unk and
  232 + # unbounded or multiple unknown terms) so if we
  233 + # weren't able to resolve the boundedness by now, that indicates a
  234 + # problem so we report the error showing the troublesome terms which are
  235 + # now in uu.
  236 + if unknown:
  237 + if bad:
  238 + msg = 'unknown and unbounded terms present in %s'
  239 + elif unknown:
  240 + msg = 'multiple terms with unknown boundedness in %s'
  241 + raise NotImplementedError(msg % uu)
  242 + # OR
  243 + # (3) the troublesome terms have been identified as finite or unbounded
  244 + # and we proceed with the non-error code since the lists have been updated.
  245 +
  246 + u = Add(*unknown_result)
  247 + if unbounded_result or unbounded_const:
  248 + unbounded.extend(zero)
  249 + inf_limit = Add(*(unbounded_result + unbounded_const))
200 250 if inf_limit is not S.NaN:
201 251 return inf_limit + u
202 252 if finite:
13 sympy/series/tests/test_limits.py
@@ -294,9 +294,12 @@ def test_extended_real_line():
294 294 assert limit(oo - x, x, -oo) == oo
295 295 assert limit(x**2/(x-5) - oo, x, oo) == -oo
296 296 assert limit(1/(x+sin(x)) - oo, x, 0) == -oo
  297 + assert limit(oo/x, x, oo) == oo
  298 +
  299 +@XFAIL
  300 +def test_extended_real_line_fail():
297 301 assert limit(x - oo + 1/x, x, oo) == -oo
298 302 assert limit(x - oo + 1/x, x, 0) == -oo
299   - assert limit(oo/x, x, oo) == oo
300 303
301 304 @XFAIL
302 305 def test_order_oo():
@@ -316,3 +319,11 @@ def test_Limit_dir():
316 319 def test_polynomial():
317 320 assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, oo) == 1
318 321 assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, -oo) == 1
  322 +
  323 +def test_issue_2641():
  324 + assert limit(log(x)/z - log(2*x)/z, x, 0) == -log(2)/z
  325 +
  326 +def test_issue_3267():
  327 + n = Symbol('n', integer=True, positive=True)
  328 + r = (n + 1)*x**(n + 1)/(x**(n + 1) - 1) - x/(x - 1)
  329 + assert limit(r, x, 1) == n/2

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