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I need help with an integral in Mathematica:

I need to calculate the integral of x^(1/4)*BesselJ[-1/4, a*x]*Cos[b*x] in the x variable (a and b are parameters) between 0 and Infinity.

The function is complicated and no analytic primitive exist, but when I tried to do it numerically with NIntegrate it did not converge. However x^(1/4)*BesselJ[-1/4, a*x] does converge (and it can be calculated analytically in fact) so the other one should converge and the problem with Mathematica must be some numerical error.

Stephen Ostermiller
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user3742054
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  • If I use a=1 and b=1 and plot your two expressions I see the Bessel without the Cos has alternating positive and negative peaks while the Bessel with the Cos has positive and roughly zero peaks. If build a table of the value of each integral from 0 to n Pi for increasing integer values of n it appears that your claim that your first integral must converge is very likely false. – Bill Jun 15 '14 at 17:20
  • You can analytically get the definite integral from {0,x1} (use `Assumptions->x1>0` ) for specific `a,b`. then convince yourself the result goes to infinity as `x1->Infinity`. – agentp Jun 16 '14 at 16:24
  • On further look, this thing seems like it maybe does converge at least for `a>b`. Unfortunately the integral only evaluates symbolically for a==b, so you are stuck looking at it numerically. For example a=2,b=1 the integral asymptotes to around 0.3 with an oscillation that seems to be decreasing in magnitude for large x1. Try taking this to math.stackexchange.com. – agentp Jun 16 '14 at 18:49

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