Addition Formulas of Leaf Functions According to Integral Root of Polynomial Based on Analogies of Inverse Trigonometric Functions and Inverse Lemniscate Functions

TL;DR

This paper introduces leaf functions derived from differential equations related to polynomial roots, extending classical trigonometric and lemniscatic functions, and presents addition formulas for these functions based on their integral roots.

Contribution

The paper develops new leaf functions for different polynomial degrees and derives addition formulas using integral root analogies of inverse functions.

Findings

Leaf functions generalize trigonometric and lemniscatic functions.

Addition formulas for leaf functions are established for n=1, 2, and 3.

Graphs illustrate the periodic waveforms of these functions.

Abstract

The second derivative of a function r(t) with respect to a variable t is equal to -n times the function raised to the 2n-1 power of r(t); using this definition, an ordinary differential equation is constructed. Graphs with the horizontal axis as the variable t and the vertical axis as the variable r(t) are created by numerically solving the ordinary differential equation. These graphs show several regular waves with a specific periodicity and waveform depending on the natural number n. In this study, the functions that satisfy the ordinary differential equation are presented as the leaf functions. For n = 1, the leaf function become a trigonometric function. For n = 2, the leaf function becomes a lemniscatic elliptic function. These functions involve an additional theorem. In this paper, based on the additional theorem for n = 1 and n = 2, the double angle and additional theorem for n = 3 are presented.