Re: Integral Fails to Converge

• To: mathgroup at smc.vnet.net
• Subject: [mg101894] Re: Integral Fails to Converge
• From: antononcube <antononcube at gmail.com>
• Date: Mon, 20 Jul 2009 19:22:58 -0400 (EDT)
• References: <h3pelq\$o36\$1@smc.vnet.net>

```Hi,

First let me point out that you probably have used FORTRAN notation to
specify numerical values.
If we assume that with (1 E-3) you mean (1/1000) and we redefine your
constants to have infinite precision (shown below) then NIntegrate has
no problem calculating the integral with t set to 1.

In[660]:= t = 1;
a = 105/10000;
cw = 454 *(10^(-12));
gammaw = 9810;
volume = 802 *(10^(-7));
e = 208 (10^8);
p = 23/100;
nu = 10^(-6);
n = 1/100;
l = 15/100;

cs = (3*(1 - (2*p)))/e;
s = l*gammaw*((n*cw) + cs);

K = 10^(-20);
T = K*l*gammaw/nu;

In[659]:= {a, cw, gammaw, volume, e, p, nu, n, l, cs, s, K, T} // N

Out[659]= {0.0105, 4.54*^-10, 9810., 0.0000802, 2.08*^10, 0.23,
1.*^-6, 0.01, 0.15, 7.788461538461539*^-11,
1.2128782153846155*^-7, 1.*^-20, 1.4715*^-11}

In[674]:= alpha = \[Pi]*(a^2)*s/(volume*cw*gammaw);
beta = \[Pi]*T*t/(volume*cw*gammaw);

In[676]:= Clear[f1]
f1[u_] := (((u*BesselJ[0, u]) -
2*alpha*BesselJ[1, u])^2) + (((u*BesselY[0, u]) -
2*alpha*BesselY[1, u])^2);

In[678]:=
f1n[u_?NumericQ] := (((u*BesselJ[0, u]) -
2*alpha*BesselJ[1, u])^2) + (((u*BesselY[0, u]) -
2*alpha*BesselY[1, u])^2);

In[679]:= f = (8*alpha/(\[Pi]^2))*
Integrate[(((Exp[-beta*u*u/alpha])/(u*f1[u]))), {u, 0, Infinity}]

Out[679]= (8860131*Integrate[1/(E^((3250000*u^2)/2953377)*
(u*((u*BesselJ[0, u] - (8860131*Pi*BesselJ[1, u])/
118335100)^2 + (u*BesselY[0, u] -
(8860131*Pi*BesselY[1, u])/118335100)^2))),
{u, 0, Infinity}])/(29583775*Pi)

In[680]:= f = (8*alpha/(\[Pi]^2))*
NIntegrate[(((Exp[-beta*u*u/alpha])/(u*f1n[u]))), {u, 0, Infinity}]

Out[680]= 0.6974195846386315

In[683]:= Plot[Re[((Exp[-beta*u*u/alpha])/(u*f1n[u]))], {u, 0, 10},
PlotRange -> All]

On Jul 17, 5:01 am, Antoine Letendre <antoine.leten... at gmail.com>
wrote:
> Hi!
>
> I have a question regarding the Integral of a complicated function.
> The program I am running is the following:
>
> t =. ..;
> a = 0.0105;
> cw = 4.54 E - 10;
> gammaw = 9.81 E + 3;
> volume = 8.02 E - 5;
> e = 2.08 E + 10;
> p = 0.23;
> nu = 1 E - 6;
> n = 0.01;
> l = 0.15;
>
> cs = (3*(1 - (2*p)))/e;
> s = l*gammaw*((n*cw) + cs);
>
> K = 1 E - 20;
> T = K*l*gammaw/nu;
>
> alpha = \[Pi]*(a^2)*s/(volume*cw*gammaw);
> beta = \[Pi]*T*t/(volume*cw*gammaw);
>
> f1 = (((u*BesselJ[0, u]) -
>        2*alpha*BesselJ[1, u])^2) + (((u*BesselY[0, u]) -
>        2*alpha*BesselY[1, u])^2);
>
> f = (8*alpha/(\[Pi]^2))*
>   Integrate[(((Exp[-beta*u*u/alpha])/(u*f1))), {u, 0, Infinity}]
>
> Unfortunately it gives the error:NIntegrate::ncvb:NIntegratefailed to con=
verge to prescribed accuracy
> after 9 recursive bisections in u near {u} = {0.000201031}.NIntegrate
> obtained
> 7.35544*10^6+3.1321576550246354049389926310603926238860307749390128833831=