Fractional Iteration via Ramanujan's Master Theorem

A uniform, unchanging density can still carry transport. The current comes from the phase gradient, and a plane wave stands for a long packet, a scattering state or a particle beam, never a single normalized particle.

Replacing dx with √dx in a Riemann sum multiplies the ordinary integral by √n. A finite answer therefore needs a zero integral and a precisely tuned singularity.

An expected dipole moment that oscillates at the transition frequency, fed into the Larmor formula. The result is close to the quantum emission rate, but a single factor shows where the classical argument breaks down.