Re: Definite Integration in Mathematica (continuous antiderivative from Gass' book)
- To: mathgroup at smc.vnet.net
- Subject: [mg74677] Re: Definite Integration in Mathematica (continuous antiderivative from Gass' book)
- From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
- Date: Sat, 31 Mar 2007 01:28:16 -0500 (EST)
- References: <eufq77$7hj$1@smc.vnet.net> <euigcj$oug$1@smc.vnet.net>
"dimitris" <dimmechan at yahoo.com> wrote:
> This question appeared in my previous post but may be to its length
> could stay unnoticed.
>
> Consider the antiderivative of 1/(2+Cos[x]).
>
> f = HoldForm[Derivative[-1][1/(Cos[#1] + 2) & ][x]]
>
> Mathematica returns an antiderivative having a jump discontinuity
> at x=Pi in the range [0,2Pi].
>
> F = ReleaseHold[f]
> TrigFactor[D[F, x]]
> Plot[F, {x, 0, 2*Pi}, Ticks -> {Range[0, 2*Pi, Pi/2], Automatic}]
>
> (2*ArcTan[Tan[x/2]/Sqrt[3]])/Sqrt[3]
> 1/(2 + Cos[x])
>
> This is normal since the integrand is not analytic in the complex
> plane. So any candidate for an antidrivative of this function, will
> have to have singularities somewhere in the complex plane.
>
> So the correct application of the Newton Leibniz formula is
>
> (F /. x -> 2*Pi) - Limit[F, x -> Pi, Direction -> -1] + Limit[F, x ->
> Pi, Direction -> 1] - (F /. x -> 0)
> (2*Pi)/Sqrt[3]
>
> and that precisely does Mathematica.
>
> Integrate[1/(2 + Cos[x]), {x, 0, 2*Pi}]
> {N[%], NIntegrate[1/(2 + Cos[x]), {x, 0, 2*Pi}]}
>
> (2*Pi)/Sqrt[3]
> {3.6275987284684352, 3.6275987284707627}
>
> One can take a continuous antiderivative in the interval
> [0,2Pi] by adding the piecewise constant function:
>
> (Limit[(2*ArcTan[Tan[x/2]/Sqrt[3]])/Sqrt[3], x -> Pi, Direction -> #1]
> & ) /@ {-1, 1}
> {-(Pi/Sqrt[3]), Pi/Sqrt[3]}
>
> Fc[x_] := Piecewise[{{F, -Pi < x < Pi}, {2*(Pi/Sqrt[3]), x == Pi}, {F
> + 2*(Pi/Sqrt[3]), Pi < x < 3*Pi}}]
>
> Plot[Fc[x], {x, 0, 2*Pi}, Ticks -> {Range[0, 2*Pi, Pi/2], Automatic}]
>
> Fc[2*Pi] - Fc[0]
> (2*Pi)/Sqrt[3]
>
> My question now...
>
> >From an old book about Mathematica (Mathematica for Scientists and
> Enginners by R. Gass we ge (but without further explanation...)
>
> FF[x_] = (-2*3^(1/2)*ArcTan[Sin[x]/(Cos[x] + 1)])/3 +
> (2*3^(1/2)*ArcTan[(3^(1/2)*Sin[x])/(3*(Cos[x] + 1))])/3 +
> (3^(1/2)*x)/3
>
> D[FF[x], x]//Simplify
> 1/(2 + Cos[x])
>
> Plot[FF[x], {x, 0, 2*Pi}, Ticks -> {Range[0, 2*Pi, Pi/2], Automatic}]
>
> as a continuous antiderivative in the real axis for the integrand.
>
> So the Newton-Leibniz formula reads
>
> FF[2*Pi] - FF[0]
> (2*Pi)/Sqrt[3]
>
> I know that my question does not have much connection with CAS issues
> but can someone explain me how exactly could we take FF[x]?
>
> Is it possible to obtain FF[x] by a CAS following some way?
I _already_ showed, in my previous post in this thread, one way to obtain
the result given by Gass. Furthermore, that method could indeed be
automated for a CAS.
Also please note (because you continue to say otherwise) that the result
given by Gass is NOT "a continuous antiderivative in the real axis for the
integrand" as far as Mathematica is concerned. That's due to his expression
being Indeterminate when x = (2n + 1)Pi, n integer.
My previous post, however, gave an antiderivative which IS valid
throughout R.
David