The Compositional Integral: A Brief Introduction

TL;DR

The paper introduces the Compositional Integral as a novel approach to integration and differential equations, generalizing Riemann's integral through infinitely nested compositions and offering a new perspective inspired by Euler's method.

Contribution

It formally defines and constructs the Compositional Integral, providing a new notation and conceptual framework for understanding first-order differential equations.

Findings

Provides a formal definition of the Compositional Integral

Connects the method to Riemann sums and Euler's method

Suggests applications for infinitely nested compositions

Abstract

The Compositional Integral is defined, formally constructed, and discussed. A direct generalization of Riemann's construction of the integral; it is intended as an alternative way of looking at First Order Differential Equations. This brief notice aims to: familiarize the reader with a different approach to integration, fabricate a notation for a modified integral, and express a startling use for infinitely nested compositions. Taking inspiration from Euler's Method for approximating First Order Differential Equations, we affiliate the method with Riemann Sums; and look at it from a different, modern angle.