The Fourier Transform of the Stretched Exponential Function: Behavior at the Center and in the Tails

The characterization of spectral lineshapes originating from relaxation processes in highly disordered systems remains a significant challenge in materials science and optical spectroscopy. For instance, the frequency-domain representation of the stretched exponential (Kohlrausch-Williams-Watts, KWW) function, sometimes mentioned for observations in complex media, lacks a closed analytical form. In this work, we propose a framework rooted in Tsallis non-extensive statistical mechanics, introducing the generalized q-Gaussian distribution as a global analytical model to describe the numerical Fourier transform of KWW relaxation functions. Using high-resolution Discrete Fourier Transform (DFT) grids, we systematically evaluated the performance of these competing lineshapes for a characteristic disorder parameter of β=0.6. Our global optimization protocol reveals that the q-Gaussian function exhibits absolute geometric precision across multiple decades of intensity, naturally converging into a distinct over-Lorentzian heavy-tailed regime (q≈2.66). Furthermore, we address and resolve the apparent mathematical paradox arising between the classical complex-plane saddle-point asymptotic expansions—which formally dictate an ultra-deep power-exponential decay—and the numerical optimization results. We will show that within the physically and computationally accessible frequency windows, the transformed KWW function is entirely governed by an extensive transient phase that acts as a highly flexible power law. The continuous q-Gaussian smoothly bridges the flat, analytic, Gaussian-like core near the origin with the highly sustained algebraic wings, offering a unified framework for spectroscopy and complex relaxation physics.