Integration of BesselJ[1,z] and BesselJ[0,z]
- To: mathgroup at smc.vnet.net
- Subject: [mg41779] Integration of BesselJ[1,z] and BesselJ[0,z]
- From: "RJM" <rmatyi at comcast.net>
- Date: Thu, 5 Jun 2003 07:31:24 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
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
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- From: Daniel Lichtblau <danl@wolfram.com>
- Re: Integration of BesselJ[1,z] and BesselJ[0,z]