Re: NIntegrate about singular point

*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 > documentation center to learn more details about the various options and > 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 - Thanks for your help. I get a lot from you mentioned.