|
|
Re: Removing singularity at x=0 for integral
Posted:
Nov 27, 2013 6:20 PM
|
|
On Tuesday, November 26, 2013 3:10:29 PM UTC-5, Axel Vogt wrote:
> On 26.11.2013 16:10, dspguy2t wrote:
>
> > Hi I'm trying to numerically integrate the following expression from 0 .. 1
>
> > f(x)=1/x^1.5 * exp(-A/x) , where A is a constant.
>
> >
>
> > As x->0, the exponential goes to zero faster than the power, so f(x)->0, as x-> 0.
>
> > I'm evaluating this integral at various values of A - there are cases when A is very small.
>
> >
>
> > I'm trying to find a way to remove the singularity at x=0. I've looked at the usual techniques, like integration by parts, subtracting the singularity out, change of variables.
>
> >
>
> > The change of variables is usually done when the power is less than one e.g.(1/sqrt(x)). exp(-A/x) is not analytic so integration by parts seems problematic.
>
> >
>
> > Any suggestions?
>
> >
>
> > Thanks for any assistance.
>
> > David
>
>
>
> Guessing you mean 0 < A the value is -Pi^(1/2)*(erf(A^(1/2))-1)/A^(1/2) by Maple
>
>
>
> Note that you do not need 'analytic', so x = 1/t^2 is allowed here and will
>
> give what Maple says for Int(2*exp(-A*t^2), t = 1 .. infinity)
Hi Axel,
Thanks for the help. Yes A>0. Given that I need to solve the integral numerically, integrating out to infinity becomes a little problematic (when A is very small). Most texts usually avoid the integration to infinity by making the reverse substitution.
I was able to make some progress by integrating by parts (see post below). The reason I'm integrating numerically is I'm also looking at a similar, but more difficult integral - which definitely can't be done in closed form.
Thanks,
David
|
|
