# A classical functional generalization of the first Barnes lemma

## TL;DR

This paper presents a simplified proof of a contour integral formula for the Gauss hypergeometric function, which generalizes Barnes's first lemma and offers an alternative to existing formulas.

## Contribution

It introduces a more straightforward proof and a new generalization of Barnes's first lemma for hypergeometric functions.

## Key findings

- Provides a simpler proof of the contour integral formula.
- Generalizes Barnes's first lemma for hypergeometric functions.
- Offers an alternative to existing integral formulas.

## Abstract

We give a brief account and a simpler proof of a contour integral formula for the Gauss hypergeometric function. Such formula is alternative to Barnes's integral formula and generalizes the first Barnes Lemma.