Re: Integration of BesselJ[1,z] and BesselJ[0,z]
- To: mathgroup at smc.vnet.net
- Subject: [mg41832] Re: Integration of BesselJ[1,z] and BesselJ[0,z]
- From: bobhanlon at aol.com (Bob Hanlon)
- Date: Fri, 6 Jun 2003 09:51:06 -0400 (EDT)
- References: <bbna3i$2ac$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Increase the precision:
Integinten[ts_] = Integrate[BesselJ[0, t0], {t0, 0, ts}];
Table[{ts, N[Integinten[ts], 25]}, {ts, 0, 50, 1/10}];
ListPlot[%, PlotRange -> All, PlotJoined -> True];
Bob Hanlon
In article <bbna3i$2ac$1 at smc.vnet.net>, "RJM" <rmatyi at comcast.net> wrote:
<< Subject: Integration of BesselJ[1,z] and BesselJ[0,z]
From: "RJM" <rmatyi at comcast.net>
To: mathgroup at smc.vnet.net
Date: Thu, 5 Jun 2003 11:38:26 +0000 (UTC)
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
>><BR><BR>