The real part of the complementary error function

TL;DR

This paper proves that the real part of the complementary error function is bounded below by 1 in a specific complex plane region, improving previous zero-free results using classical estimates and complex analysis.

Contribution

It introduces a new proof for the lower bound of the real part of the complementary error function in a complex domain, utilizing an old estimate and elementary complex analysis.

Findings

The real part of the complementary error function is ≥ 1 in a specific complex region.

The proof employs a classical estimate for the error function from nearly 60 years ago.

The result enhances understanding of the error function's behavior in the complex plane.

Abstract

It is shown that the real part of the complementary error function is bounded below by 1 in the subset of the complex plane where the principal argument is between $3\pi/4$ and $5\pi/4$. This improves a previous result asserting that the complementary error function has no zeros in the same set, whose proof was essentially based on calculus of two variables. The present proof uses an almost $60$ years old, relatively unknown estimate for the error function, combined with some elementary complex analysis, thus serving as a humble illustration to a quote attributed to Jacques Hadamard: "The shortest path between two truths in the real domain passes through the complex domain".