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Re: Bug in Mathematica or my mistake?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89159] Re: Bug in Mathematica or my mistake?
  • From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Tue, 27 May 2008 07:16:17 -0400 (EDT)
  • References: <g1avld$fb5$1@smc.vnet.net> <g1dhvb$9n6$1@smc.vnet.net>

"ram.rachum at gmail.com" <ram.rachum at gmail.com> wrote:
> On May 25, 9:06 am, "ram.rac... at gmail.com" <ram.rac... at gmail.com>
> wrote:
> > I am trying to find the limit of this expression:
> >
> > (a t Sqrt[1 - (a^2 t^2)/c^2] +
> >    c ArcSin[(a t)/
> >      c])/(2 a (-(v Sqrt[(c^2 - v^2)/c^2] + c ArcSin[v/c])/(2
> > \!\(\*OverscriptBox["a", "0"]\)) + 1/(2
> > \!\(\*OverscriptBox["a",
> >        "0"]\))(c ArcSin[(
> >          v + (c^2 - a t v -
> >            Sqrt[(c^2 - a t v)^2 + a t^2 (-2 (c^2 + a d) + a^2 t^2)
> > \!\(\*OverscriptBox["a", "0"]\)] Sign[a t
> > \!\(\*OverscriptBox["a", "0"]\)])/(a t))/
> >          c] + (v + (
> >           c^2 - a t v -
> >            Sqrt[(c^2 - a t v)^2 + a t^2 (-2 (c^2 + a d) + a^2 t^2)
> > \!\(\*OverscriptBox["a", "0"]\)] Sign[a t
> > \!\(\*OverscriptBox["a", "0"]\)])/(a t)) Sqrt[(
> >         c^2 - (v + (
> >            c^2 - a t v -
> >             Sqrt[(c^2 - a t v)^2 + a t^2 (-2 (c^2 + a d) + a^2 t^2)
> > \!\(\*OverscriptBox["a", "0"]\)] Sign[a t
> > \!\(\*OverscriptBox["a", "0"]\)])/(a t))^2)/c^2])))
> >
> > The limit calculation takes a lot of time, and then it says zero. But
> > when I put in some numbers with "/.", and plot it, it converges not to
> > zero but to other values, like 1 or 1.5, depending on the numbers I
> > put in. So what's going on? How can Mathematica tell me that the limit
> > is zero in the general case, but not zero in a specific case?

It's a bug, and should be reported.

> I apologize: I forgot to include some vital information. The limit I'm
> taking is with t->0.
>
> Also, you might want to know the values I assigned to the other
> variables when I did the numeric computation. They are these:
>
> {c -> 300000, a -> 10,
> \!\(\*OverscriptBox["a", "0"]\) -> 1, v -> 7, d -> 10}
>
> Although other values will work too, this is just an example.
>
> Another thing is when you plot it, you need to pump up the
> WorkingPrecision. A hundred was enough for my example. You can do the
> plot as t goes from 0 to 10, for example, and see how it converges to
> a non-zero value.

I suspect that your desired limit is

1/( (1 + a d/c^2) Sqrt[1 - (v/c)^2] )

There might be an easy way to get that result from Mathematica, but I don't
know what that way would be.

Note that the above is independent of \!\(\*OverscriptBox["a", "0"]\).

David W. Cantrell


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