The fundamental theorem of simple harmonic functions

Abstract:

The defining relation for a one-dimensional real-valued simple harmonic function designated as$\ cine$ (pronounced “scene” and symbolized $cin\left (x\right )$) is introduced as $\frac {d}{dx}cin\left (x\right )=cin\left (x+p\right )$. It is theorized that this definition of the derivative as the function itself translated on the real line by a certain positive number $p$ is a sufficient condition for proving that $cin\left (x\right )$ also satisfies the one-dimensional oscillator equation $\frac {d^2}{dx^2}cin\left (x\right )=-\ cin\left (x\right )$. A proof of this hypothesis is provided by establishing the continuity, differential, and boundedness properties of $cin\left (x\right )$ and all of its higher-order derivatives and antiderivatives, while relying critically on Roe’s characterization of the sine function. The $cine$ function enables the adoption of independent non-circular definitions of the trigonometric functions $sine$ and $cosine$. The properties of $cine$ are investigated, and trigonometric symmetries and identities are derived directly from the defining relation and its corollaries. A formulation for the unique solution of the function is proposed. Several useful operational theorems are stated and proved. The function is applied to solve problems in trigonometry and physics.