A visual and simple proof for the Picard Little Theorem

TL;DR

This paper presents a concise, visual proof of the Little Picard Theorem in complex analysis, relying solely on basic concepts and geometric intuition, making it accessible for standard courses.

Contribution

It introduces a new, simple, and visual proof of the Little Picard Theorem using only fundamental concepts, avoiding complex technical tools.

Findings

Provides a short, visual proof of the Little Picard Theorem

Uses only basic concepts from standard complex analysis courses

Highlights geometric properties of the exponential map

Abstract

One of the most famous results in Complex Analysis is the Little Picard Theorem, that characterizes the image set of an arbitrary entire function. Specifically, the theorem states that this image set is either the whole complex plane or the whole complex plane except a point. The traditional proofs for the theorem involve technical tools such as either modular functions, Harnack's inequality, Bloch and Landau theorems... The previous fact makes the Little Picard Theorem to be often presented at initial courses on Complex Analysis without a proof. This manuscript provides a short and visual proof by only using basic concepts that are covered in any standard course on Complex Analysis. In fact, the essence of the proof is a good understanding of composition of the complex exponential map with itself and its underlying geometrical properties.