Melin transform of the floor function [x]

[Rfael]

Homework Statement

using analytic continuation can i compute the mellin transform of the floor function as a riemann zeta function

Relevant Equations

$$ \int_{0}^{\infty}[x]x^{s-1}= - \frac{\zeta (-s)}{s} $$

usig analytic continuation and mellin transform properties

 

[pasmith]

How do you define $[z]$ for $z \in \mathbb{C}$?

 

[renormalize]

Relevant Equations: $$ \int_{0}^{\infty}[x]x^{s-1}= - \frac{\zeta (-s)}{s} $$

Just a small quibble: although the meaning of your integral is clear from the context of your equation, please don't forget to always explicitly include the differential of the variable you're integrating over:$$\int_{0}^{\infty}[x]x^{s-1}dx$$

 

[julian]

Your question could have been phrased more clearly.

Try evaluating your integral by first expressing it as $\sum_{n=0}^\infty \int_n^{n+1} [x] x^{s-1} dx$ and then writing out the resulting series. That should make the connection to $\zeta (-s)$ clear—if that's what you're aiming for.