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MathGroup Archive 2007

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Re: Solving a Integral (2)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg78124] Re: Solving a Integral (2)
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Sat, 23 Jun 2007 07:13:50 -0400 (EDT)
  • References: <f5g9bn$g6b$1@smc.vnet.net>

Hi again.

I try various things in order to help Integrate.
With no success unfortunately.

Say

In[2]:=
f[x_, a_, b_, c_, d_] = x^(c - 1)*(Exp[-(x/d)]/(1 + Exp[(-a)*x - b]))

Out[2]=
x^(-1 + c)/(E^(x/d)*(1 + E^(-b - a*x)))

(*your integrand*)

The requested definite integral stays unevaluated.

In[9]:=
Integrate[f[x, a, b, c, d], {x, 0, Infinity}]

Out[9]=
Integrate[x^(-1 + c)/(E^(x/d)*(1 + E^(-b - a*x))), {x, 0, Infinity}]

This is not surprising. The integrand contains four parameters
and it is not an easy task to deal with all of them.

If specific values are given; say

In[11]:=
ex = Thread[{a, b, c, d} -> {3, 2, 1, 4}]

Out[11]=
{a -> 3, b -> 2, c -> 1, d -> 4}

then the task is much more easier!

In[33]:=
FullSimplify[Integrate[f[x, a, b, c, d] /. ex, {x, 0, Infinity}]]

Out[33]=
(1/3)*(-1)^(1/12)*E^(1/6)*Log[((1 + 2/(-1 +
(-1)^(1/4)*E^(1/6)))^(-1)^(1/6)*(1 + 2/(-1 +
(-1)^(5/12)*E^(1/6)))^(-1)^(1/3)*
     (1 + 2/(-1 + (-1)^(7/12)*E^(1/6)))^I*(1 + 2/(-1 +
(-1)^(3/4)*E^(1/6)))^(-1)^(2/3)*
     (1 + 2/(-1 + (-1)^(11/12)*E^(1/6)))^(-1)^(5/6)*(-(-1)^(11/12) +
E^(1/6)))/((-1)^(11/12) + E^(1/6))]

In[34]:=
{N[%], NIntegrate[f[x, a, b, c, d] /. ex, {x, 0, Infinity}]}

Out[34]=
{3.961045612743195 - 5.551115123125783*^-16*I, 3.9610456087403803}

Let's try to leave to leave one parameter unspecified; say d.

In[42]:=
Integrate[f[x, 2, 1/2, 3, d], {x, 0, Infinity}]

Out[42]=
If[Re[d] > 0, 2*d^3*(1 - HypergeometricPFQ[{1, 1 + 1/(2*d), 1 + 1/
(2*d), 1 + 1/(2*d)}, {2 + 1/(2*d), 2 + 1/(2*d), 2 + 1/(2*d)},
      -(1/Sqrt[E])]/((1 + 2*d)^3*Sqrt[E])), Integrate[x^2/(E^(x/d)*(1
+ E^(-(1/2) - 2*x))), {x, 0, Infinity},
   Assumptions -> Re[d] <= 0]]

Say now we leave two parameters unspecified.

In[47]:=
Integrate[f[x, a, 5/3, 1, d], {x, 0, Infinity}]

Out[47]=
If[Re[a] < 0 && Re[a] < Re[1/d], ((-1)^(1/(a*d))*E^(5/(3*a*d))*Beta[-
E^(5/3), 1 - 1/(a*d), 0])/a,
  Integrate[1/(E^(x/d)*(1 + E^(-(5/3) - a*x))), {x, 0, Infinity},
Assumptions -> Re[a] >= Re[1/d] || Re[a] >= 0]]

Dimitris

PS

See this message as a hint; not as the final word!







ehrnsperge... at pg.com       :
> Jean-Marc,
>
> thanks for your help. I made the changes as you suggested and I still can
> not convince Mathematica to solve the integral. Any additional
> suggestions?
>
> Thanks,
>
> Bruno
>
> Dr. Bruno Ehrnsperger
> Principal Scientist
>
> Procter & Gamble Service GmbH
> Sulzbacherstr.40
> 65824 Schwalbach
> Germany
>
> fon +49-6196-89-4412
> fax +49-6196-89-22965
> e-mail: ehrnsperger.b at pg.com
> internet: www.pg.com
>
> Gesch=E4ftsf=FChrer: Otmar W. Debald, Gerhard Ritter, Dr. Klaus Schumann,
> Willi Schwerdtle
> Sitz: Sulzbacher Str. 40, 65824 Schwalbach am Taunus, Amtsgericht:
> K=F6nigstein im Taunus HRB 4990
>
>
>
>
>
> Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
> 21/06/2007 15:30
>
>
>         To:     Bruno Ehrnsperger-B/PGI@PGI, mathgroup at smc.vnet.net
>         cc:
>         Subject:        Re: Solving a Integral
>
>
> ehrnsperger.b at pg.com wrote:
> > I need help in solving the following integral:
> >
> > Integral = 1/(beta^alpha* Gamma[alpha]) *
> > Integrate[x^(alpha-1)*Exp[-x/beta]/(1+Exp[-a*x-b]),{x,0, infinity},
> -----------------------------------------------------------^^^^^^^^
> oo Must be written Infinity (with a capital I)
>
> > Assumptions: (alpha> 0)||(beta > 0)||(a > 0)||(b <0)]
> -------------^^
> The : character means nothing here: use ->
>
> Moreover, are you sure that you want a OR ( that is ||) condition on
> your assumptions rather than an AND (that is &&)?
>
> HTH,
> Jean-Marc
>
> > The Integral is approximately 1/(beta^alpha* Gamma[alpha])
> > *1/(1+Exp[-a*alpha*beta-b]) + Order[alpha*beta^2]
> >
> > However, I would like to have an exact analytical solution, and I am
> > failing to convince Mathematica to give me the solution. Is there a way
> to
> > ask Mathematica to give the solution as a series expansion of my
> > approximate solution?
> >
> > Thanks so much for your help,
> >
> > Bruno
> >
> > Dr. Bruno Ehrnsperger
> > Principal Scientist
> >
> > Procter & Gamble Service GmbH
> > Sulzbacherstr.40
> > 65824 Schwalbach
> > Germany
> >
> > fon +49-6196-89-4412
> > fax +49-6196-89-22965
> > e-mail: ehrnsperger.b at pg.com
> > internet: www.pg.com
> >
> > Gesch=E4ftsf=FChrer: Otmar W. Debald, Gerhard Ritter, Dr. Klaus
> Schumann,
> > Willi Schwerdtle
> > Sitz: Sulzbacher Str. 40, 65824 Schwalbach am Taunus, Amtsgericht:
> > K=F6nigstein im Taunus HRB 4990
> >
> >



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