Analytical and Numerical Study of the Zeros of the Error Function, Local Conformality and Sectorial Asymptotic Behavior

Abstract

The complex error function erf(z) plays a critical role in applied mathematics, physics and engineering; however, several of its deeper analytical properties in the complex domain remain insufficiently explored. This paper presents a comprehensive analytical and numerical study of erf(z), focusing on its zero distribution, conformal mapping behavior and sectorial asymptotic structure. By employing classical tools from complex analysis, including Rouché's Theorem, the Argument Principle and contour integration, we investigate the structure of potential zero-free regions and provide numerical evidence for the absence of zeros in certain bounded sectors. Specifically, numerical exploration suggests that no zeros occur in selected bounded angular sectors within a computational precision of 10^(-8), consistent with the analytical structure derived in this study. The conformal character of erf(z), as an entire and non-constant function, is examined through Jacobian analysis and domain-coloring visualizations, revealing its angle-preserving yet scale-distorting transformations of canonical domains. Furthermore, we analyze sectorial asymptotic expansions as |z|→∞ and demonstrate that Padé approximants reduce the relative numerical error by up to 30% compared with classical truncated series, particularly in boundary sectors where convergence deteriorates. These results provide a more robust understanding of the analytic and geometric behavior of erf(z), with practical implications for complex modeling in statistical physics, quantum theory and signal processing.