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