05/07/2014, 03:22 PM
Well I posted a reply to this but it deleted it :/
I have looked at this for so long Tommy. It's very closely related to tetration. I'll tell you how I can do it for some functions.
\( M(f) = \int_0^\infty f(x)x^{s-1}\,dx \)
define
\( \vartheta(w) = \sum_{n=0}^\infty M^n(f)(s) \frac{w^n}{n!} \)
Then if
\( \phi(z) = [\frac{d^z}{dw^z} \vartheta(w)]_{w=0} \)
then
\( \phi(z) = M^z (f)(s) \)
However we have to show lots of conditions on convergence and what not.
I have looked at this for so long Tommy. It's very closely related to tetration. I'll tell you how I can do it for some functions.
\( M(f) = \int_0^\infty f(x)x^{s-1}\,dx \)
define
\( \vartheta(w) = \sum_{n=0}^\infty M^n(f)(s) \frac{w^n}{n!} \)
Then if
\( \phi(z) = [\frac{d^z}{dw^z} \vartheta(w)]_{w=0} \)
then
\( \phi(z) = M^z (f)(s) \)
However we have to show lots of conditions on convergence and what not.