       A NIntegrate question

• To: mathgroup at smc.vnet.net
• Subject: [mg52029] A NIntegrate question
• From: acsl at dee.ufrj.br (Antonio Carlos Siqueira)
• Date: Sun, 7 Nov 2004 01:04:44 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Dear All,
I need to find the Inverse Fourier Transform of an expression
involving Sqrt, Exp and Bessel Functions. As my interest is in the
numerical response I am trying to use NIntegrate for frequency to time
transform. In fact I am using it to have a Fourier Cossine transform.
My problem is that I can only find an answer if I use interpolation
for the frequency domain function. I was wondering if anybody can give
me a help in trying to use NIntegrate to solve this problem. What I
have tried didn´t work that well. Any comments are welcome.
Regards
Antonio
Here comes my functions....

zint[s_, rhoc_:8.82573*10^-8, rc_:0.0203454, mu_:4.0 Pi 10^-7] :=
Sqrt[s mu/rhoc] rhoc/(2 Pi rc)BesselI[0, Sqrt[s mu/rhoc]
rc]/BesselI[
1, Sqrt[s mu/rhoc] rc]

Ze[s_,mu_:4.0 Pi 10^-7,
rc_:0.0203454,rhoc_:8.82573*10^-8,rhosolo_:100.0] :=
s mu/(2 Pi) Log[2*(15.0+ Sqrt[rhosolo/(s mu)]/rc]

Zserie[s_]:=Ze[s]+zint[s];

Y[s_,e_:1/(36.0 Pi 10^9)]:=s 2 Pi e /(Log[2*15.0/rf])

A[s_]:=Exp[Sqrt[Zserie[s]*Y[s]]*-10000.0]
(* from the graphic one can see that this funciton is bounded and goes
to zero as s goes to either I*Infinity or -I*Infinity *)

a[t_]:=2/Pi NIntegrate[Im[A[I 2 Pi x]/(2 Pi x) Cos[ 2 Pi x t],
{x,0,\[Infinity]}, MaxRecursion -> 50, Method->Oscillatory]

The error message did not help much for my case

"Numerical integration stopping due to loss of precision.
Achieved neither the requested PrecisionGoal nor AccuracyGoal;
suspect one of the following: highly oscillatory integrand or
the true value of the integral is 0. If your integrand is oscillatory
try
using the option Method->Oscillatory in NIntegrate.

```

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