Peskin Eq 11.72, mathematical identity

[Hao]

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.

 

[vela]

$$\partial_\alpha x^\alpha = \partial_\alpha e^{\alpha \log x} = e^{\alpha \log x} \log x= x^\alpha \log x$$

 

[Hao]

Awesome!

Thanks!