Re: A Problem with Simplify
- To: mathgroup at smc.vnet.net
- Subject: [mg87760] Re: A Problem with Simplify
- From: Alexey Popkov <popkov at gmail.com>
- Date: Wed, 16 Apr 2008 07:14:00 -0400 (EDT)
- References: <ftkb7f$a9m$1@smc.vnet.net> <200804140943.FAA08090@smc.vnet.net>
On 16 =C1=D0=D2, 13:06, Daniel Lichtblau <d... at wolfram.com> wrote: > Alexey Popkov wrote: > > On 15 =C1=D0=D2, 13:51, Daniel Lichtblau <d... at wolfram.com> wrote:= > > >>Alexey Popkov wrote: > > >>>On Apr 10, 10:14 am, "Kevin J. McCann" <Kevin.McC... at umbc.edu> wrote: > > >>>>I have the following rather simple integral of two sines, which should= > >>>>evaluate to zero if m is not equal to n and to L/2 if they are the sam= e.= > > >>>Try the following: > >>>Integrate[Exp[(a - 1)*x], x] /. a -> 1 > >>>Integrate[Cos[(a - 1)*x], x] /. a -> 1 > >>>Integrate[(a - 1)^x, {x, -1, 0}] /. a -> 1 > >>>Integrate[Cos[a x]/Sin[x], x] /. a -> 1 > > >>>There is the ONE underlying BUG! In some complicated cases this bug > >>>may result in random partial answers. > > >>>http://forum.ru-board.com/topic.cgi?forum=5&topic=10291&start=80#= 9 > > >>Removing the replacements, here are the Integrate results. > > >>In[17]:= InputForm[Integrate[Exp[(a - 1)*x], x]] > >>Out[17]//InputForm= E^((-1 + a)*x)/(-1 + a) > >> [...] > >>As far as I am aware thse are correct. What is the bug? > > >>Daniel Lichtblau > >>Wolfram Research > > > The above answers are correct only if "a" is not equal to 1. Answers > > with a=1 are lost! > > > And code: > > > int1 = Integrate[Cos[a*x]/Sin[x], x]; > > int1 /. a -> 1 > > > MUST give us answer equal to > > > int2 = Integrate[Cos[x]/Sin[x], x] > > Capitalization notwithstanding, it need not do any such thing. Integrate > should give a generic result that behaves sensibly in the limit as a->1 > from whatever path is specified to Limit. This is similar to the > situation of > > In[18]:= InputForm[solns = x /. Solve[a*x^2 + x == 1, x]] > Out[18]//InputForm= {(-1 - Sqrt[1 + 4*a])/(2*a), > (-1 + Sqrt[1 + 4*a])/(2*a)} > > In[20]:= Limit[solns, a->0] > Out[20]= {-Infinity, 1} > > Note that a blind substitution of a->0 will not give "nice" results. > > In[21]:= solns /. a->0 > > 1 > Power::infy: Infinite expression - encountered. > 0 > > 1 > Power::infy: Infinite expression - encountered. > 0 > > Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered. > > Out[21]= {ComplexInfinity, Indeterminate} > > > But the first answer is invalid in this simple case... :( > > > I should emphasize that in some complicated cases this BUG may result > > in random partial answers! > > I'll emphasize that this is not a bug; it's the way Integrate is > intended to work in Mathematica. An antiderivative of a function that > depends on parameters can be invalid for some values of those > parameters. Provided we are talking about analytic functions (things > that can be said to have antiderivatives in the usual meaning of the > term), the "bad set" will have measure zero. > > Daniel Lichtblau > Wolfram Research >Capitalization notwithstanding, it need not do any such thing. Integrate >should give a generic result that behaves sensibly in the limit as a->1 >from whatever path is specified to Limit. This is similar to the >situation of >In[18]:= InputForm[solns = x /. Solve[a*x^2 + x == 1, x]] >Out[18]//InputForm= {(-1 - Sqrt[1 + 4*a])/(2*a), > (-1 + Sqrt[1 + 4*a])/(2*a)} > >In[20]:= Limit[solns, a->0] >Out[20]= {-Infinity, 1} > >Note that a blind substitution of a->0 will not give "nice" results. Well. But how I can get the partial answer for the case a=1 with Limit? I have tried the foolowing code: In[1]:= Limit[Integrate[Cos[a*z]/Sin[z],z],a->1] Out[1]= \[Infinity] In[2]:= Limit[Integrate[Exp[a*z]*Sinh[z],z],a->1] Out[2]= -\[Infinity] In[3]:= Limit[Integrate[Exp[z]*Sinh[a z],z],a->1] Out[3]= E^z DirectedInfinity[E^(-I*Im[z])] In[4]:= Limit[Integrate[Exp[(a-1)*x],x],a->1] Out[4]= \[Infinity] All results are incorrect! As I understand, Integrate does not give a generic result that behaves sensibly in the limit as a->1!
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- Re: A Problem with Simplify
- From: Alexey Popkov <popkov@gmail.com>
- Re: A Problem with Simplify