# The Lemniscatic Functions

## TL;DR

This paper develops the mathematical theory of lemniscatic functions sl and cl, exploring their properties as solutions to a specific initial value problem, contributing to the understanding of these special functions.

## Contribution

It introduces a formal development of lemniscatic functions from their initial value problem, expanding the theoretical framework of these functions.

## Key findings

- Defined the lemniscatic functions sl and cl as solutions to an initial value problem.
- Analyzed properties and behaviors of these functions.
- Established foundational results for further mathematical exploration.

## Abstract

We develop the theory of the lemniscatic functions sl and cl from their definition as solutions to an initial value problem.