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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 �A G[1] is singular point�AP[1]=1
> and I try Plot the graph for G[a]�AP[a]�Aand G[a]*P[a] �Afrom 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>