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Numerical Integration
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71471] Numerical Integration
*From*: "dimitris" <dimmechan at yahoo.com>
*Date*: Mon, 20 Nov 2006 02:43:53 -0500 (EST)
Dear All,
I have one question about the numerical integration of one function.
$VersionNumber
5.2
Firstly, consider the following two functions
f[x_] := Tan[Cos[x]*Exp[-x/10]]
g[x_] := Tan[BesselJ[0, x]*Exp[-x/30]]
Plot[{f[x], g[x]}, {x, 0, 30}, PlotPoints -> 100, Axes -> None, Frame
-> {True, True, False, False}, PlotStyle -> {Red, Blue}]
MapThread[{#1, Limit[{f[x], h[x]}, x -> #1, Direction -> #2]} & , {{0,
Infinity}, {-1, 1}}]
{{0, {Tan[1], Tan[1]}}, {Infinity, {0, 0}}}
Here are numerical estimations of their integral over {0,Infinity}
Timing[NIntegrate[f[x], {x, 0, Infinity}, MaxRecursion -> 16,
PrecisionGoal -> 20,
WorkingPrecision -> 40, SingularityDepth -> 1000]]
{0.6090000000000089*Second,
0.23162115314242937349322778892580847913`20.06942527752895}
Timing[NIntegrate[g[x], {x, 0, Infinity}, MaxRecursion -> 16,
PrecisionGoal -> 20,
WorkingPrecision -> 40, SingularityDepth -> 1000]]
{4.891000000000002*Second,
1.42946176273365811506679399034735564921`20.05898967415425}
Now consider the following function
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
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