Re: Integration of BesselJ[1,z] and BesselJ[0,z]
- To: mathgroup at smc.vnet.net
- Subject: [mg41809] Re: Integration of BesselJ[1,z] and BesselJ[0,z]
- From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
- Date: Fri, 6 Jun 2003 09:50:39 -0400 (EDT)
- Organization: Universitaet Leipzig
- References: <bbna3i$2ac$1@smc.vnet.net>
- Reply-to: kuska at informatik.uni-leipzig.de
- Sender: owner-wri-mathgroup at wolfram.com
Hi,
that is an error in the approximation of the StruveH[]
function, you can see this with
Plot[StruveH[1, ts], {ts, 20, 30}]
and
Plot[StruveH[0, ts], {ts, 26, 29}]
because your integration result:
test = Integrate[BesselJ[0, t0], {t0, 0, ts}] // FunctionExpand
include the StruveH[] functions.
Regards
Jens
RJM wrote:
>
> Hello,
>
> I am having problems with the integration of the Bessel function of the
> first kind. If I use the expression for the first order function
> (BesselJ[1,z]), the function itself is just fine, showing the expected
> damped oscillatory behavior starting at (x,y) = (0,0). When integrated
> (Integrate [BesselJ[1, t1], {t1, 0, t}]) the result is again the expected
> result with a damped oscillation converging to +1. However, if I do the same
> using the zero-order function BesselJ[0,z] the starting function again looks
> fine starting at (x,y) = (0,1) with damped oscillations -- but when I
> integrate BesselJ[0,z], the result starts to get "noisy" after the fifth
> local maximum, very noisy 6th local maximum, junping to y=0 at the 6th local
> minimum. After the noisy 7th local maximum, however, the integral "settles
> down" to the expected damped oscillation converging on +1!! The code to
> generate plots showing this behavior is as follows:
>
> << Graphics`Graphics`
>
> Table[{e1, BesselJ[1, e1]}, {e1, 0, 50, 0.2}];
> ListPlot[%, PlotRange -> All, PlotJoined -> True]
>
> Integinten[t_] = Integrate [BesselJ[1, t1], {t1, 0, t}];
> Table[{t, Integinten[t]}, {t, 0, 50, 0.1}];
> ListPlot[%, PlotRange -> All, PlotJoined -> True]
>
> Table[{e0, BesselJ[0, e0]}, {e0, 0, 50, 0.2}];
> ListPlot[%, PlotRange -> All, PlotJoined -> True]
>
> Integinten[ts_] = Integrate [BesselJ[0, t0], {t0, 0, ts}];
> Table[{ts, Integinten[ts]}, {ts, 0, 50, 0.1}];
> ListPlot[%, PlotRange -> All, PlotJoined -> True]
>
> I have run this using version 4.0 under Windows 98 and version 4.2.1 under
> Windows 2000 with nominally identical results. Any explanations on this
> strange behavior or a proposed fix would be appreciated.
>
> Regards,
> Rich Matyi