In the pole expansion of gamma function,
I hope to differentiate the pole part and the regular part with respect to $\alpha$ and to see the sum of the two derivatives vanishes. This shows that the right hand side does not depend on the particular choice of $\alpha$, which conventionally is set to 1.
Do I have to differentiate the intergal? Also, isn't $\alpha$ 1 already in the pole expansion. How does it appear?