About using series function

Question

I am using Series function on (1/z)(-k^2)^z. Up to z^0, the function gives me 1/z + Log[-k^2]. But in the standard textbook on QFT, it turns out the expansion should give 1/z + Log[k^2/mu^2] -i*pi up to z^0 in the series. Could you explain the reason behind such discrepancy? How to resolve it?

Answer 1

There is probably a constant $\mu$ present in your Quantum Field Theory book.

For the rest, analytically, your function has definition

$$\frac{1}{z}\left( -k^2\right)^z= e^{z \log(-k^2) - \log z} = e^{2 z \log k\ \pm i \ \pi \ z - \log z}$$