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Peskin Eq 11.72, mathematical identity

  1. Feb 7, 2010 #1

    Hao

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    In Eq 11.72 in the QFT text by Peskin, the following equality is stated:

    [tex]i\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\log(k_{E}^{2}+m^{2})=-i\frac{\partial}{\partial\alpha}\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}[/tex]

    This suggests that

    [tex]\log(k_{E}^{2}+m^{2})=-\frac{\partial}{\partial\alpha}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}[/tex]

    However, I can't see how this identity follows. Differentiating the right hand side gives

    [tex]-\frac{\partial}{\partial\alpha}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}=\frac{\alpha}{(k_{E}^{2}+m^{2})^{\alpha+1}}|_{\alpha=0}\rightarrow\frac{0}{(k_{E}^{2}+m^{2})^{1}}[/tex]

    Any help would be greatly appreciated.
     
  2. jcsd
  3. Feb 7, 2010 #2

    vela

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    [tex]\partial_\alpha x^\alpha = \partial_\alpha e^{\alpha \log x} = e^{\alpha \log x} \log x= x^\alpha \log x[/tex]
     
  4. Feb 7, 2010 #3

    Hao

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    Awesome!

    Thanks!
     
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