The Imani Periodic Functions: Genesis and Preliminary Results

TL;DR

This paper introduces the Imani functions, a new class of periodic solutions derived from a specific Hamiltonian, which share properties with classical trigonometric functions, and discusses their preliminary properties and unresolved questions.

Contribution

The paper defines and constructs the Imani functions, a novel class of periodic solutions related to a specific Hamiltonian, expanding the understanding of functional equations and special functions.

Findings

Imani functions are periodic solutions with properties similar to sine and cosine.

Explicit construction of Imani functions is provided.

Several unresolved issues regarding Imani functions are identified.

Abstract

The Leah-Hamiltonian, $H(x,y)=y^2/2+3x^{4/3}/4$, is introduced as a functional equation for $x(t)$ and $y(t)$. By means of a nonlinear transformation to new independent variables, we show that this functional equation has a special class of periodic solutions which we designate the Imani functions. The explicit construction of these functions is done such that they possess many of the general properties of the standard trigonometric cosine and sine functions. We conclude by providing a listing of a number of currently unresolved issues relating to the Imani functions.