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Integration of x^n*e^x/(e^x + 1)^2

  1. Apr 12, 2010 #1
    I there.

    I'm currently using this kind of integrals, with n even, and I couldn't found anything in internet for calculate this.
    Let


    From the book I'm studying Ashcroft/Mermin, Solid State Physics, Append C, it says that
    [tex]a_{n}=\int_{-\infty }^{\infty } \frac{x^n
    e^x}{\left(e^x+1\right)^2} \, dx[/tex]

    can, by elementary operations, be written as

    [tex]a_{n}=1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}-\frac{1}{4^{2n}}+...[/tex] and so can be written with the zeta function:

    [tex]a_{n}=(2-\frac{1}{2^{2(n+1)}}) \zeta(2n)[/tex]

    and, [tex]\zeta(2n)=2^{2n-1}\frac{\pi^{2n}}{(2n)!}B_{n}[/tex] where B_n are the bernoulli numbers.

    Well, aren't any easier way, using integration in complex plane? Can you give me an ideia of where can I find a resolution to this?

    Thanks,
    littlepig
     
  2. jcsd
  3. Apr 12, 2010 #2
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