  We investigate the propagation characteristics of Cauchy-Riemann beams in
gradient-index media. Our study reveals two key findings: a) the preservation
of their form during propagation, and b) surprisingly, the feasibility of
obtaining the fractional Fourier transform for any arbitrary entire function.
We provide an explicit and straightforward expression for this transform.
Additionally, our results contribute to a deeper understanding of the behavior
of Cauchy-Riemann beams in complex media, offering insights that may have
implications for various applications in optics and wave propagation.
