Journal of Applied and Pure Mathematics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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ON THE COMPLEX-VALUED DEGENERATE LAMBERT W FUNCTION AND ITS APPLICATIONS

  • SUNG-SOO PYO (Department of Mathematics Education, Silla University)
  • Received : 2025.09.17
  • Accepted : 2026.02.08
  • Published : 2026.03.30
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Abstract

This study establishes the analytical extension of the complexvalued degenerate Lambert W function, Wλ(z), and provides a rigorous characterization of its multi-valued structure. A primary theoretical contribution of this work is the formal proof demonstrating that Wλ(z) is a non-elementary function, established through its independence from finite compositions of basic algebraic and transcendental functions. Building upon this framework, we derive an explicit closed-form root formula for the trinomial-type equation xλ + ax + b = 0, offering a precise analytical solution for a broad class of nonlinear algebraic problems. Furthermore, we demonstrate the utility of the Wλ function by constructing exact solutions to specific categories of first-order nonlinear differential equations, thereby expanding the existing analytical toolkit for nonlinear mathematical modeling.

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