Hao
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In Eq 11.72 in the QFT text by Peskin, the following equality is stated:
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}
This suggests that
\log(k_{E}^{2}+m^{2})=-\frac{\partial}{\partial\alpha}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}
However, I can't see how this identity follows. Differentiating the right hand side gives
-\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}}
Any help would be greatly appreciated.
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}
This suggests that
\log(k_{E}^{2}+m^{2})=-\frac{\partial}{\partial\alpha}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0}
However, I can't see how this identity follows. Differentiating the right hand side gives
-\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}}
Any help would be greatly appreciated.