08/22/2013, 05:54 PM
Let's take some function \( f(t) \) and some fractional differentiation method \( \frac{d^s}{dt^s} \) such that \( \frac{d^s f}{dt^s}(-t) < e^{-t} \)
Now create the function:
\( \phi(s) = \int_0^\infty t^{s-1} \frac{d^s f}{dt^s}(-t) dt \)
Integrate by parts, and for \( \Re(s) > 0 \) we get the spectacular identity that:
\( s \phi(s) = \phi(s+1) \)
What is going on here?
Now create the function:
\( \phi(s) = \int_0^\infty t^{s-1} \frac{d^s f}{dt^s}(-t) dt \)
Integrate by parts, and for \( \Re(s) > 0 \) we get the spectacular identity that:
\( s \phi(s) = \phi(s+1) \)
What is going on here?