I cannot understand an integration in the proof of the Direct mapping Theorem for the Mellin transform. A statement of the Theorem, together with an outline of the standard proof, can be found at page 15 of this book.
After decomposing the Mellin transform into a sum, the following integral, which the author declares to be "easily computable", has to be examined:
$\int_0^1 \sum_{k,b} c_{k,b} x^{s+b-1} log(x)^k dx, \qquad b\in\mathbb{R}, k\in\mathbb{N},$
And $s\in\mathbb{C}$ is bounded from below, but in principle it could be $\Re(s)<-b$. (And $\Re(s)<-b$ indeed happens, for example applying this Theorem to obtain a meromorphic extension for the Gamma function.)
Integrating by parts I proved:
$\int_0^1 x^rlog(x)^n = \frac{(-1)^n n!}{(r+1)^{n+1}}, \qquad \forall \:r\in\mathbb{R}_{\geq -1},\:n\in\mathbb{N}.$
But it seems to me that he claims this result to hold for any $r\in\mathbb{R}$.
Question Do you see a gap in my arguments? Or do you know where to find a detailed proof of the Direct mapping Theorem for Mellin transform?
Thank you very much!