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MathGroup Archive 2006

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oscillatory integrals

  • To: mathgroup at
  • Subject: [mg70070] oscillatory integrals
  • From: dimmechan at
  • 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

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

5.2 for Microsoft Windows (June 20, 2005)

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





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.


The option Method->Oscillatory will be employed


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

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

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


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:


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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