a definite integral

• To: mathgroup at smc.vnet.net
• Subject: [mg77083] a definite integral
• From: dimitris <dimmechan at yahoo.com>
• Date: Sun, 3 Jun 2007 06:07:00 -0400 (EDT)

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]=
{5.99068,7.64822,8.52201,6.97126,5.02596,7.44021,9.8316,5.03409,8.95359,1.\
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

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