An Explanation of Mellin's 1921 Paper

TL;DR

This paper explains Mellin's 1921 method for computing the principal root of a polynomial using hypergeometric functions and integral transforms, making the original complex analysis accessible to a broader audience.

Contribution

It provides an accessible exposition of Mellin's original work, clarifying his use of hypergeometric functions and integral transforms for polynomial root computation.

Findings

Mellin's method computes polynomial roots using hypergeometric functions.

The paper clarifies Mellin's integral transform approach.

It makes complex analysis techniques more accessible to non-experts.

Abstract

In 1921 Mellin published a Comptes Rendu paper computing the principal root of the polynomial $Z^n + x_1Z^{n_1} + \cdots + x_pZ^{n_p} - 1$ using hypergeometric functions of its coefficients $x_1,...,x_p.$ He used an integral transform nowadays bearing his name. Slightly over three pages, the paper is written in French in a terse style befitting the language. Unable to find an elementary explanation on the web or in a texbook, we wrote this expository article to make Mellin's landmark result accessible to interested people who are not experts in hypergeometric functions and complex analysis.