#2412 Problems with integral (x/(exp(x)+1),x,0,inf)

None
closed
nobody
5
2013-01-03
2012-05-25
No

The correct result of integrate(x/(exp(x)+1),x,0,inf); should be pi^2/12.

Maxima returns:
(%o2) limit(-x*log(%e^x+1)-li[2](-%e^x)+x^2/2,x,inf,minus)-%pi^2/12
build_info("5.27.0","2012-04-24 08:52:03","i686-pc-mingw32","GNU Common Lisp (GCL)","GCL 2.6.8")

regards

chris

Discussion

  • Aleksas

    Aleksas - 2012-05-29

    my solution:

    (%i1) S:'integrate(x/(exp(x)+1),x,0,inf)$

    (%i2) intparts(S,u):=block([f,var,v,a,b],
    f:part(S,1),var:part(S,2),v:integrate(f/u,var),
    if last(S)#var then
    (a:part(S,3),b:part(S,4),
    limit(u*v,var,b,minus)-limit(u*v,var,a,plus)
    -'integrate(v*diff(u,var),var,a,b))
    else u*v-'integrate(v*diff(u,var),var)
    )$

    (%i3) intparts(S,1/(exp(x)+1));
    (%o3) integrate((x^2*%e^x)/(%e^x+1)^2,x,0,inf)/2
    (%i4) ev(%, nouns);
    (%o4) %pi^2/12

    (%i5) float(%), numer;
    (%o5) 0.82246703342411
    (%i6) quad_qagi(x/(%e^x+1), x, 0, inf);
    (%o6) [0.82246703342411,5.9689712369638092*10^-10,135,0]

    Aleksas D

     
  • christoph reineke

    Thank you very much!
    By the way, we have the same problem with the similar expression
    integrate(x/(exp(x)-1),x,0,inf)=pi^2/6.
    Sorry, I noticed that after submitting my first message!

    Regards

    Chris

     
  • Aleksas

    Aleksas - 2012-05-29

    Solution of the second problem:
    (%i1) S:'integrate(x/(exp(x)-1),x,0,inf);
    (%o1) integrate(x/(%e^x-1),x,0,inf)

    (%i2) intparts(S,u):=block([f,var,v,a,b],
    f:part(S,1),var:part(S,2),v:integrate(f/u,var),
    if last(S)#var then
    (a:part(S,3),b:part(S,4),
    limit(u*v,var,b,minus)-limit(u*v,var,a,plus)
    -'integrate(v*diff(u,var),var,a,b))
    else u*v-'integrate(v*diff(u,var),var)
    )$

    (%i3) changevar(S, y=exp(x)-1, y, x);
    (%o3) integrate(log(y+1)/(y^2+y),y,0,inf)

    (%i4) intparts(%,log(y+1));
    (%o4) -integrate((log(y)-log(y+1))/(y+1),y,0,inf)

    (%i5) logcontract(%);
    (%o5) -integrate(log(y/(y+1))/(y+1),y,0,inf)

    (%i6) ev(%, nouns);
    (%o6) %pi^2/6

    Aleksas D

     
  • christoph reineke

    Yes, it works. Thanks again!
    Shall we close this bug report?

    Regards

    Chris

     
  • Dan Gildea

    Dan Gildea - 2013-01-03

    added asymptotic expansion for polylogarithm - can compute limit now.

     
  • Dan Gildea

    Dan Gildea - 2013-01-03
    • status: open --> closed
    • milestone: --> None
     

Log in to post a comment.

Get latest updates about Open Source Projects, Conferences and News.

Sign up for the SourceForge newsletter:





No, thanks