       • To: mathgroup at smc.vnet.net
• Subject: [mg126118] Re: NIntegrate about singular point
• From: bowlderster <bowlderster at gmail.com>
• Date: Thu, 19 Apr 2012 03:53:22 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jmjf8p\$89v\$1@smc.vnet.net> <jmls0d\$jdp\$1@smc.vnet.net>

```On 4=D4=C218=C8=D5, =CF=C2=CE=E73=CA=B154=B7=D6, A Retey <a... at gmx-topmail.de> wrote:
> Hi,
>
>
>
>
>
> > I am dealing with an integral as following
>
> > h = 1
> > m = 1
> > n = 1
> > t = 1
> > k = 4.0269
> > kk = 4.0284
> > Plot[x^(m + n - 1)/(x*Sinh[x*h] -
> >       kk*Cosh[x*h])*((x +
> >         kk)*((-1)^(m + n)*Exp[x*(2.*(-0.2) + h)] - ((-1)^m + (-1)^n)*
> >          Exp[-x*h]) + (x - kk)*Exp[-x*(2.*(-0.2) + h)]), {x, 0, 40}]
> > NIntegrate[
> >   x^(m + n - 1)/(x*Sinh[x*h] -
> >       kk*Cosh[x*h])*((x +
> >         kk)*((-1)^(m + n)*Exp[x*(2.*(-0.2) + h)] - ((-1)^m + (-1)^n)*
> >          Exp[-x*h]) + (x - kk)*Exp[-x*(2.*(-0.2) + h)]), {x, 0,
> >    Infinity}]
>
> > It has a singular point when the denominator is zero.
> > At begining, I try to solve it in another system, yet with Nan result.
>
> > It is the first time for me to use Mathematica.
> > Can it solve the integral with singular point?
>
> I think that NIntegrate handles many singularities automatically if the
> integral actually converges. I haven't checked, but I suspect that in
> this case the integral is probably simply not converging. It's easy
> enough though to get the cauchy principle value, if that is what you're
> after:
>
> expr = x^(m + n - 1)/(x*Sinh[x*h] -
>       kk*Cosh[x*h])*((x +
>         kk)*((-1)^(m + n)*Exp[x*(2.*(-0.2) + h)] - ((-1)^m + (-1)^n)*
>          Exp[-x*h]) + (x - kk)*Exp[-x*(2.*(-0.2) + h)]);
>
> singularity = x /. FindRoot[Denominator[expr] == 0, {x, 4}]
>
> NIntegrate[expr, {x, 0, Infinity}, Exclusions -> {singularity},
>   Method -> "PrincipalValue"]
>
> you might want to have a look at tutorial/NIntegrateOverview in the
> methods of NIntegrate.
>
> hth,
>
> albert- =D2=FE=B2=D8=B1=BB=D2=FD=D3=C3=CE=C4=D7=D6 -
>
> - =CF=D4=CA=BE=D2=FD=D3=C3=B5=C4=CE=C4=D7=D6 -