A visual and simple proof for the Picard Little Theorem
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.
Cite
@article{arxiv.2308.06159,
title = {A visual and simple proof for the Picard Little Theorem},
author = {Daniel Cao Labora},
journal= {arXiv preprint arXiv:2308.06159},
year = {2023}
}
Comments
There is an error when claiming that "a continuity argument obviously extends this property to any z such that Re(f(z)) in (0,1/2)" page 10 (it is not obvious why this could be true). Some attempts have been done in order to solve this issue in a future version, but unsuccesful until now. If fixed, I will submit a new version