04/05/2023, 04:56 AM
It looks like proving the connection between fractional integrals and these residues is actually quite easy. Take
\[ \int_{-\infty}^\infty f(x^2) dx\]
Then by a change of variables this becomes
\[\int_{0}^\infty \frac{f(t)}{\sqrt{t}} dt \]
Which is nothing more than the half order fractional derivative (after diving by gamma) evaluated at zero, exactly as predicted. So for instance, this shows that the results I've derive for \( e^{-x^2} \) are valid. This also solves that \( \eta \) case illustrated before, and says that its value is just the half order fractional derivative.
\[ \int_{-\infty}^\infty f(x^2) dx\]
Then by a change of variables this becomes
\[\int_{0}^\infty \frac{f(t)}{\sqrt{t}} dt \]
Which is nothing more than the half order fractional derivative (after diving by gamma) evaluated at zero, exactly as predicted. So for instance, this shows that the results I've derive for \( e^{-x^2} \) are valid. This also solves that \( \eta \) case illustrated before, and says that its value is just the half order fractional derivative.