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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

• Follow-Ups:
  • Re: Integration of BesselJ[1,z] and BesselJ[0,z]
    • From: Daniel Lichtblau <danl@wolfram.com>