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Re: Numerical Integration


dimitris schrieb:
> Dear All,
> 
> I have one question about the numerical integration of one function.
> 
> $VersionNumber
> 5.2
> 
...
> h[x_] := Tan[BesselJ[0, x]]
> 
> Plot[h[x], {x, 0, 40}, PlotPoints -> 100, Axes -> None, Frame -> {True,
> True, False, False}, PlotStyle -> AbsoluteThickness[2]]
> 
> Limit[h[x], x -> Infinity]
> 0
> 
> I try hard to find any proper settings for getting a numerical
> estimation of its integral
> over {0,Infinity} but I can't succeed.
> 
> Any help will be greatly appreciate.
> 
> Dimitris
> 

Hi Dimitris,

I tried it this way:

In[1]:=
Needs["NumericalMath`BesselZeros`"];
h[x_] := Tan[BesselJ[0, x]];
t0 = SessionTime[];
bzlist = NestList[BesselJZerosInterval[0, {1, 2}*Last[#1] + {-1/10, 1/10}] & ,
    Flatten[{0, BesselJZeros[0, 2]}], 9];
v0 = (NIntegrate[h[x], Evaluate[Flatten[{x, #1}]]] & ) /@ bzlist;
SequenceLimit[Rest[FoldList[Plus, 0, v0]]]
(SessionTime[] - t0)*seconds
Out[6]=
1.4545133229307878
Out[7]=
1.75*seconds

The displayed result (1.45451) does not change any more when increasing the
number of intervals from 9 to 10 or more.

Peter


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