Re: Integration of Singular function

• To: mathgroup at smc.vnet.net
• Subject: [mg80015] Re: Integration of Singular function
• From: antononcube <antononcube at gmail.com>
• Date: Fri, 10 Aug 2007 06:42:03 -0400 (EDT)
• References: <f9elv3\$jiv\$1@smc.vnet.net>

```In version 6.0 you can use the method AdaptiveMonteCarlo:

In[183]:= Clear[rf1, rf2];

In[184]:=
rf1[x_, y_] := (0.0666667 \[ExponentialE]^-((-1 + x)^2 + (-1 +
y)^2)/(4 t)
Im[((1 + 0.1516 \[ImaginaryI]) (1 + x + \[ImaginaryI] y) ((

1 + x + \[ImaginaryI] y)/(-1 + x + \[ImaginaryI] y))^(
0.0758 \[ImaginaryI]))/(-1 + (x + \[ImaginaryI] y)^2)^(3/
2)])/t

In[185]:=
rf2[x_, y_] := (0.621099 (\[ExponentialE]^-((-1 + x)^2 + (-1 +
y)^2)/(4 t) -
1/
3 \[ExponentialE]^-((-1 + x)^2 + (1 + y)^2)/(4 t)) Im[((1 +
0.1516 \[ImaginaryI]) (1 + x + \[ImaginaryI] y) ((

1 + x + \[ImaginaryI] y)/(-1 + x + \[ImaginaryI] y))^(
0.0758 \[ImaginaryI]))/(-1 + (x + \[ImaginaryI] y)^2)^(3/
2)])/t

In[190]:= II = Table[{t,
NIntegrate[

rf1[x, y], {y, -\[Infinity],
0}, {x, -\[Infinity], \[Infinity]},
MaxRecursion -> 60, Method -> AdaptiveMonteCarlo,
MaxPoints -> 500000000] +
NIntegrate[

rf2[x, y], {y,
0, \[Infinity]}, {x, -\[Infinity], \[Infinity]},
MaxRecursion -> 60, Method -> AdaptiveMonteCarlo,
MaxPoints -> 500000000]}, {t, 0.0001, 5, 0.2}]; // Timing

Out[190]= {21.4577, Null}

On Aug 9, 4:15 am, Khandelwal <ratne... at gmail.com> wrote:
> Hello,
>
> I am having trouble integrating the following functions in
> Mathematica
>
> II=Table[{t,
>  NIntegrate[
>     rf1[x, y], {y, -\[Infinity], 0}, {x, -\[Infinity], \[Infinity]},
>     MaxRecursion -> 60, Method -> MonteCarlo[24],
>     MaxPoints -> 500000000] +
>    NIntegrate[
>     rf2[x, y], {y, 0, \[Infinity]}, {x, -\[Infinity], \[Infinity]},
>     MaxRecursion -> 60, Method -> MonteCarlo[24],
>     MaxPoints -> 500000000]}, {t, 0.0001, 5, 0.2}]
>
> rf1[x_,y_]=(0.0666667 \[ExponentialE]^-((-1 + x)^2 + (-1 + y)^2)/(4 t)
>    Im[((1 + 0.1516 \[ImaginaryI]) (1 + x + \[ImaginaryI] y) ((
>     1 + x + \[ImaginaryI] y)/(-1 + x + \[ImaginaryI] y))^(
>    0.0758 \[ImaginaryI]))/(-1 + (x + \[ImaginaryI] y)^2)^(3/2)])/t
>
> rf1[x_,y_]=(0.621099 (\[ExponentialE]^-((-1 + x)^2 + (-1 + y)^2)/(4 t) -
>    1/3 \[ExponentialE]^-((-1 + x)^2 + (1 + y)^2)/(4 t)) Im[((1 +
>      0.1516 \[ImaginaryI]) (1 + x + \[ImaginaryI] y) ((
>     1 + x + \[ImaginaryI] y)/(-1 + x + \[ImaginaryI] y))^(
>    0.0758 \[ImaginaryI]))/(-1 + (x + \[ImaginaryI] y)^2)^(3/2)])/t
>
> My problem is though i'm taking too large value of MaxPoints
> (MaxPoints -> 500000000), because of that it's takes lot of time, but
> still for some inital small value of t(0.0001) its not conversing. I
> want to plot (t,II), where t->(0,5). Just wondering if there is other
> better way of dealing this integration!!
>
> --
> Regards,
> Ratnesh Khandelwal
> IISc,Bangalore,INDIA

```

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