Re: a definite integral
- To: mathgroup at smc.vnet.net
- Subject: [mg77117] Re: [mg77083] a definite integral
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Mon, 4 Jun 2007 03:48:39 -0400 (EDT)
- References: <200706031007.GAA02666@smc.vnet.net>
On 3 Jun 2007, at 19:07, dimitris wrote: > dint=Integrate[BesselJ[0, x]BesselJ[0, t - x], {x, 0, t}] > > Mathematica (5.2) returns unevaluated the integral. > > However the result is actually Sin[t]. > > Check > > In[77]:= > o = Table[Random[Real, {1, 10}], {10}] > Out[77]= > {5.99068,7.64822,8.52201,6.97126,5.02596,7.44021,9.8316,5.03409,8.9535 > 9,1.\ > 34601} > > In[78]:= > (NIntegrate[BesselJ[0, u]*BesselJ[0, #1 - u], {u, 0, #1}] & ) /@ o > Out[78]= > {-0.288349,0.978906,0.785043,0.635047,-0.951238,0.915609,-0.395691,-0. > 948699,\ > 0.453948,0.974842} > > In[79]:= > Sin[o] > Out[79]= > {-0.288349,0.978906,0.785043,0.635047,-0.951238,0.915609,-0.395691,-0. > 948699,\ > 0.453948,0.974842} > > Any ideas to show dint is equal to sin(t)? > By hand this can be proved by the convolution theorem > of the LaplaceTransform. > > Dimitris > > It's certainly not a proof but: Integrate[BesselJ[0, x]*BesselJ[0, t - x], {x, 0, t}] + O[t]^10 SeriesData[t, 0, {1, 0, -1/6, 0, 1/120, 0, -1/5040, 0, 1/362880}, 1, 10, 1] Sin[t] + O[t]^10 SeriesData[t, 0, {1, 0, -1/6, 0, 1/120, 0, -1/5040, 0, 1/362880}, 1, 10, 1] etc. Andrzej Kozlowski
- References:
- a definite integral
- From: dimitris <dimmechan@yahoo.com>
- a definite integral