oscillatory integrals

• To: mathgroup at smc.vnet.net
• Subject: [mg70070] oscillatory integrals
• From: dimmechan at yahoo.com
• Date: Mon, 2 Oct 2006 00:33:36 -0400 (EDT)

```Dear all,

I have posted this message again but since there were some
answers covering part of my queries I post it again in more readable
format.

Let me consider the integral of the function q(x,r) (see below) over
the range {x,0,Infinity}, for various values of r.

\$Version
5.2 for Microsoft Windows (June 20, 2005)
Quit

The function q(x,r) is defined as follows:

q[r_,x_]:=(f[x]/g[x])*BesselJ[0,r*x]

where

f[x_]:=x*Sqrt[x^2+1/3]*(2*x^2*Exp[(-1/5)*Sqrt[x^2+1]]-(2x^2+1)Exp[(­-1/5)*Sqrt[(x^2+1/3)]])

g[x_]:=(2x^2+1)^2-4x^2*Sqrt[x^2+1/3]Sqrt[x^2+1]

The case r=2 was first considered by Longman on a well celebrated
paper (Longman 1956).  I will consider first this case also.
>From the following plot one can see that the integrand is an
oscillatory function convergent to zero for large arguments.

In[22]:=Plot[q[2,x],{x,0,20},PlotPoints\[Rule]1000]

The option Method->Oscillatory will be employed

NIntegrate[q[2,x],{x,0,8},Method\[Rule]Oscillatory,WorkingPrecision­\[Rule]22]//Timing

Infinity::indet: Indeterminate expression 0\Infinity encountered.
Infinity::indet: Indeterminate expression 0\Infinity encountered.
{1.187 Second,-0.026608998128}

I do not understand why exist here the warning messages
Infinity::indet.

Note that despite the presence of the message, the result is very
accurate.

Now I want to plot the function NIntegrate[q[r,x],{x,0,Infinity}] in
the range {r,0,3}. What is the more reliable method to follow to get
what I want?  I simply executed

Plot[NIntegrate[q[r,x],{x,0,8},Method\[Rule]Oscillatory,
WorkingPrecision\[Rule]30],{r,0,3}]

but although I got a plot, I need considerable time and there were a
lot of warning messages so I believe this is not the case here.

Next consider the function h[r,x] which is defined as follows:

h[r_,x_]:=(f[x]/g[x])*BesselJ[1,r*x]

I want also here the plot of NIntegrate[h[r,x],{x,0,Infinity}] in
the range {r,0,3}. Because of BesselJ[1,0]=0, I am a little worry
how I will treat the point r=0.

Any suggestions? Must I check for absolute convergence?

Thanks in advance for any assistance.

Dimitrios Anagnostou

```

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