Functions and Analysis

    

The Pole Preservation Theorem for Reduced Rational Functions: A Pedagogical and Analytic Perspective.

Authors: Marciano L. Legarde

This paper introduces and formalizes the Zero-Preservation Theorem, an elementary yet structurally significant property of reduced rational functions. The theorem states that for any reduced rational function, the set of vertical asymptotes of R is identical to that of its derivative R'(x). Furthermore, the horizontal asymptote is preserved under differentiation if and only if deg P > deg Q. Proofs are provided using elementary calculus, supported by examples and counterexamples. The theorem is then shown to be a special case of a broader result from complex analysis: differentiation of meromorphic functions preserves the location of poles while increasing their order. This connection situates the theorem as a simplified corollary of a fundamental analytic principle, bridging introductory calculus concepts with the theory of meromorphic functions, pole classification, and Laurent series expansions.

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[v1] 2025-08-13 18:03:19

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