Re: Numerical integration
- To: mathgroup at smc.vnet.net
- Subject: [mg78809] Re: Numerical integration
- From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
- Date: Tue, 10 Jul 2007 06:29:12 -0400 (EDT)
- References: <f6sh65$7ob$1@smc.vnet.net>
José_Carlos_Santos <jcsantos at fc.up.pt> wrote:
> Hi all:
>
> I detected a problem concerning numerical integration
It's not a problem concerning _numerical_ integration!
NIntegrate[h[t], {t, 0, 2Pi}] would have given you a correct numerical
result.
> under Mathematica 5.1. Consider this function:
>
> h[t_] := Exp[I t](Exp[I t] - 1)Sec[Exp[I t] - 1]
>
> If I compute:
>
> N[Integrate[h[t], {t, 0, 2Pi}]]
>
> I get -3.58642.
Yes, and that is incorrect, and the reason it's incorrect is that
the _symbolic_ integral Integrate[h[t], {t, 0, 2Pi}] gives, incorrectly,
-(-2 + Pi) Pi.
The correct value of the symbolic integral is -Pi^2, which is approximately
what you got below. (BTW, can someone get version 5.x or 6 to give -Pi^2
easily?)
David
> But if I compute
>
> N[Integrate[Re[h[t]], {t, 0, 2Pi}] + Integrate[Im[h[t]], {t, 0, 2Pi}]I]
>
> instead of getting the same answer, I get -9.869604401106042 +
> 2.2377264147293624^(-16)I. Of course, the imaginary part doesn't bother
> me, but why are the results so different? BTW the difference is equal
> to 2*pi.
>
> Best regards,
>
> Jose Carlos Santos