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MathGroup Archive 2006

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Re: Hadamard Finite Part

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71516] Re: Hadamard Finite Part
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Tue, 21 Nov 2006 07:05:13 -0500 (EST)
  • References: <ejmkl6$gf5$1@smc.vnet.net>

Thanks (once more time!) to Maxim I realized where was my mistake.

Integrate[f[x]*Exp[(-e)*x], {x, 0, Infinity}, Assumptions -> e > 0]
Normal[Series[%, {e, 0, 3}]]
List @@ %
(Limit[#1, e -> 0] & ) /@ %
DeleteCases[%, _DirectedInfinity]
Plus @@ %
N[%]

(3/4)*(Zeta[5, e/2] - Zeta[5, (1 + e)/2])
24/e^5 + (31*e*Pi^6)/252 + (127*e^3*Pi^8)/1440 - (45*Zeta[5])/2 -
(2835/8)*e^2*Zeta[7]
{24/e^5, (31*e*Pi^6)/252, (127*e^3*Pi^8)/1440, -((45*Zeta[5])/2),
(-(2835/8))*e^2*Zeta[7]}
{Infinity, 0, 0, -((45*Zeta[5])/2), 0}
{0, 0, -((45*Zeta[5])/2), 0}
-((45*Zeta[5])/2)
-23.33087449072582

Also, as he suggest me, even the setting GenerateConditions->False can
give the
result -((45*Zeta[5])/2) if we simply transform the integrand so that
the singularity is at zero:

-Integrate[f[x]*dx /. x -> 1/x /. dx -> D[1/x, x], {x, 0, Infinity},
GenerateConditions -> False]
N[%]

-((45*Zeta[5])/2)
-23.33087449072582

Dimitris

dimitris wrote:
> $VersionNumber
> 5.2
>
> Consider the following function
>
> f=x^4/(1+Exp[-x]);
>
> The integral of f over {0,Infinity} is divergent
>
> Block[{Message}, Integrate[f, {x, 0, Infinity}]]
> Infinity
>
> Here are some attempts to get the finite part of the integral in the
> Hadamard sense.
>
> First with the setting GenerateConditions->False
>
> Integrate[f, {x, 0, Infinity}, GenerateConditions -> False]
> 0
>
> Then using the following setting
>
> List @@ Integrate[f, {x, 0, e}]
> (Limit[#1, e -> Infinity] & ) /@ %
> DeleteCases[%, _DirectedInfinity][[1]]
> N[%]
> {e^4*Log[1 + E^e], 4*e^3*PolyLog[2, -E^e], -12*e^2*PolyLog[3, -E^e],
> 24*e*PolyLog[4, -E^e], -24*PolyLog[5, -E^e], -((45*Zeta[5])/2)}
> {Infinity, -Infinity, Infinity, -Infinity, Infinity, -((45*Zeta[5])/2)}
> -((45*Zeta[5])/2)
> -23.33087449072582
>
> Directing removing the divergent term
>
> Integrate[f - x^4, {x, 0, Infinity}]
> N[%]
> NIntegrate[f - x^4, {x, 0, Infinity}]
> -((45*Zeta[5])/2)
> -23.33087449072582
> -23.330874489932825
>
> Using the zeta function regularization technique
>
> Integrate[fx, 0, Infinity}, GenerateConditions -> False]
> % /. s -> 4
> N[%]
> (-(-2)^(-s))*(-1 + 2^s)*Gamma[1 + s]*Zeta[1 + s]
> -((45*Zeta[5])/2)
> -23.33087449072582
>
> So the finite part of the integral is -((45*Zeta[5])/2) and not 0 as
> GenerateConditions->True
> setting might cheat us.
>
> However using a convergence implying factor Exp[-e x] I got the
> following
>
> Integrate[f*Exp[(-e)*x], {x, 0, Infinity}, Assumptions -> e > 0]
> (Limit[#1, e -> 0] & ) /@ List @@ Expand[FunctionExpand[%]]
> DeleteCases[%, _DirectedInfinity][[1]]
> N[%]
> (3/4)*(Zeta[5, e/2] - Zeta[5, (1 + e)/2])
> {(1/32)*PolyGamma[4, 1/2], Infinity}
> (1/32)*PolyGamma[4, 1/2]
> -24.108570307083355
>
> Integrate[f*Exp[(-e)*x], {x, 0, Infinity}, Assumptions -> e > 0]
> (Limit[#1, e -> 0] & ) /@ List @@ %
> DeleteCases[%, _DirectedInfinity][[1]]
> N[%]
> (3/4)*(Zeta[5, e/2] - Zeta[5, (1 + e)/2])
> {3/4, Infinity}
> 3/4
> 0.75
> 
> What I miss here?
> 
> Thanks a lot
> Dimitris


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