Chirpons: elementary local phase oscillations associated with complex zeros
For a complex wave, the local phase variation provides the instantaneous frequency in time, the local wavenumber in space, and the local momentum in quantum mechanics. Here we study this quantity from the complex-zero structure of the analytic continuation of the wave. A single nonreal zero produces a localized Lorentzian phase variation on the real axis, whose position, width, sign, amplitude, and integrated phase are fixed by the zero itself. We call this elementary zero-associated oscillation a chirpon. The representation is exact for finite products and finite Fourier sums, and extends to entire and band-limited functions through canonical factorization. We illustrate the construction with superoscillatory sequences and discrete Fourier sums, showing how local anomalous oscillations can be reconstructed from periodic families of complex zeros. We further show how linear systems transform chirpons through the displacement of their associated complex zeros, and illustrate this dynamics for quadratic dispersive propagation and a resonant harmonic oscillator. This establishes a direct relation between local phase structure and the amplitude and phase response of the transfer function, suggesting a zero-based form of spectroscopy in which system parameters can be inferred from the motion, width, and orientation of chirpons. More broadly, the chirpon description provides a unified framework for instantaneous frequency, local wavenumber, superoscillation, backflow, and other local phase phenomena, while offering a new route for probing linear systems through the dynamics of their complex zeros.