What If I told you I can find infinite functions that equal their own derivative?
Take some fractional differentiation method \( \frac{d^t}{ds^t}f(s) \) which differentiates f across s, t times. Now assume that:
\( \frac{d^t}{ds^t} f(s) < e^{-t^2} \) for some s in some set \( D \subset \mathbb{C} \), which can be easily constructed using some theorems I have.
Then:
\( \phi(s) = \int_{-\infty}^{\infty} \cos(2 \pi t) \frac{d^{t}}{ds^{t}} f(s) dt \)
If you differentiate \( \phi \) by the continuity of this improper integral \( \frac{d}{ds} \phi(s) = \phi(s) \)
What does this mean? How did I get this? Where is the mistake?
Take some fractional differentiation method \( \frac{d^t}{ds^t}f(s) \) which differentiates f across s, t times. Now assume that:
\( \frac{d^t}{ds^t} f(s) < e^{-t^2} \) for some s in some set \( D \subset \mathbb{C} \), which can be easily constructed using some theorems I have.
Then:
\( \phi(s) = \int_{-\infty}^{\infty} \cos(2 \pi t) \frac{d^{t}}{ds^{t}} f(s) dt \)
If you differentiate \( \phi \) by the continuity of this improper integral \( \frac{d}{ds} \phi(s) = \phi(s) \)
What does this mean? How did I get this? Where is the mistake?