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Algebraic Independence of an Airy Function, Its Derivative, and Antiderivative

Folkmar Bornemann

Abstract

Using tools from the Siegel-Shidlovskii theory of transcendental numbers, we prove that a nontrivial solution of the Airy equation, its derivative, and an antiderivative are algebraically independent over the field of rational functions. Courtesy of Michael Singer, the result is also derived from general considerations in differential Galois theory.

Algebraic Independence of an Airy Function, Its Derivative, and Antiderivative

Abstract

Using tools from the Siegel-Shidlovskii theory of transcendental numbers, we prove that a nontrivial solution of the Airy equation, its derivative, and an antiderivative are algebraically independent over the field of rational functions. Courtesy of Michael Singer, the result is also derived from general considerations in differential Galois theory.

Paper Structure

This paper contains 1 section, 5 theorems, 30 equations.

Table of Contents

  1. Proof of Theorem \ref{['thm']} based on differential Galois theory

Key Result

Theorem 1

Let $u\neq 0$ be a solution of $u"=z u$. Then, the functions $u, u', U$, where $U$ satisfies $U'=u$, are algebraically independent over ${\mathbb C}(z)$.

Theorems & Definitions (10)

  • Theorem 1
  • Remark
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['thm']}
  • Theorem 2