# Addition formulas of Leaf Functions and Hyperbolic Leaf Functions

## TL;DR

This paper introduces addition formulas for leaf functions and hyperbolic leaf functions, establishing their relationships and deriving key formulas like double-angle and half-angle formulas through these addition formulas.

## Contribution

The study defines hyperbolic leaf functions via imaginary numbers and derives their addition, double-angle, and half-angle formulas based on leaf functions.

## Key findings

- Derived addition formulas for hyperbolic leaf functions.
- Established relationships between leaf and hyperbolic leaf functions.
- Validated formulas through graphs and numerical data.

## Abstract

Addition formulas exist in trigonometric functions. Double-angle and half-angle formulas can be derived from these formulas. Moreover, the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number. The inverse hyperbolic function $\mathrm{arsinh}(r)=\int_{0}^{r} \frac{1}{\sqrt{1+t^2} }\mathrm{d}t$ is similar to the inverse trigonometric function $\mathrm{arcsin}(r)=\int_{0}^{r} \frac{1}{\sqrt{1-t^2} }\mathrm{d}t$, such as the second degree of a polynomial and the constant term 1, except for the sign $-$ and $+$. Such an analogy holds not only when the degree of the polynomial is 2, but also for higher degrees. As such, a function exists with respect to the leaf function through the imaginary number $i$, such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number. In this study, we refer to this function as the hyperbolic leaf function. By making such a definition, the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas, such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions. Using the addition formulas, we can also derive the double angle and half-angle formulas. We then verify the consistency of these formulas by constructing graphs and numerical data.