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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