Re: (revision) NIntegrate that upper limit is infinite

*To*: mathgroup at smc.vnet.net*Subject*: [mg71952] Re: [mg71930] (revision) NIntegrate that upper limit is infinite*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Tue, 5 Dec 2006 06:04:57 -0500 (EST)*References*: <200612041139.GAA02940@smc.vnet.net>

Evanescence wrote: > Dear all: > I am very sorry for my article contain a lot of mistake before. > So I check it and state my questions again. > I am really sorry that cause a lot of confuse for everybody. > My questions is as follows: > First I definite a function that is: > G[a_] := ((-(a -Cos[0.5])^(-3/2))/(3*(10)^2))*(Sin[0.5])^(2) + > (1/(1-0.1*0.1))*((0.005*(1 + Cos[0.5]) + 0.001/1000)*(1 + > Cos[0.5])^(1/2)*(ArcTan[(a - Cos[0.5])^(1/2)/(1 + Cos[0.5])^(1/2)] - > Pi/2) + (0.005*(1 - Cos[0.5]) - 0.001/1000)*(1 - > Cos[0.5])^(1/2)*(ArcTanh[(1 - Cos[0.5])^(1/2)/(a - Cos[0.5])^(1/2)])) > > another function is: > P[a_]:=Re[N[LegendreP[(-1/2)+I,a]]] where I=(-1)^(1/2) > > then > NIntegrate[G[a]*P[a], {a, 1, Infinity}] > > but get error message is: > NIntegrate's singularity handling has failed at point > {a}={2.7013362243395366*(10^150)}for the specified > precision goal. > Try using larger values for any of $MaxExtraPrecision > or the options WorkingPrecision, or SingularityDepth and MaxRecursion. > > NIntegrate::"inum": > "Integrand G[a]*P[a] is not numerical at {a}= > {2.7013362243395366`*(10^150)}. > > Please solve my confuse ,thank you very much! > > By the way > G[Infinity]*P[Infinity]=0 ï¿½A G[1] is singular pointï¿½AP[1]=1 > and I try Plot the graph for G[a]ï¿½AP[a]ï¿½Aand G[a]*P[a] ï¿½Afrom the > graph of G[a]*P[a] > it should convergence for a-->Infinity. I don't think this converges. gg = (100*(ArcTanh[Sqrt[1 - Cos[1/2]]/Sqrt[a - Cos[1/2]]]* (-1/1000000 + (1 - Cos[1/2])/200)*Sqrt[1 - Cos[1/2]] + (-Pi/2 + ArcTan[Sqrt[a - Cos[1/2]]/Sqrt[1 + Cos[1/2]]])* Sqrt[1 + Cos[1/2]]*(1/1000000 + (1 + Cos[1/2])/200)))/99 - Sin[1/2]^2/(300*(a - Cos[1/2])^(3/2)); pp = LegendreP[(-1/2)+I,a]; Now we check the leading terms of the series expansion at infinity. We ignore Re[]. Probably does not matter because it appears to be real valued. Normal[Series[gg*pp, {a,Infinity,1}]] ((2^(1/2 + I)*(-1667/330000 + (4999*Sqrt[-((-1 + E^(I/2))^2*(1 - Cos[1/2]))/(2*E^(I/2))])/990000 - (10001*Cos[1/2])/990000 - (Sqrt[-((-1 + E^(I/2))^2*(1 - Cos[1/2]))/(2*E^(I/2))]*Cos[1/2])/198 - Cos[1/2]^2/198)*Gamma[-2*I])/(a*Gamma[1/2 - I]^2) + (2^(1/2 - I)*(-1667/330000 + (4999*Sqrt[-((-1 + E^(I/2))^2*(1 - Cos[1/2]))/(2*E^(I/2))])/990000 - (10001*Cos[1/2])/990000 - (Sqrt[-((-1 + E^(I/2))^2*(1 - Cos[1/2]))/(2*E^(I/2))]*Cos[1/2])/198 - Cos[1/2]^2/198)*Gamma[2*I])/(a^(1 - 2*I)*Gamma[1/2 + I]^2))/a^I This looks like it goes as 1/a in magnitude. There is an oscillatory factor from imaginary powers of a, but too slow (I think) to allow for convergence. Daniel Lichtblau Wolfram Research

**References**:**(revision) NIntegrate that upper limit is infinite***From:*"Evanescence" <origine26@yahoo.com.tw>