The fundamental theorem of simple harmonic functions

Abstract

The defining relation for a one-dimensional real-valued simple harmonic function designated as c i n e \ cine (pronounced “scene” and symbolized c i n ( x ) cin\left (x\right ) ) is introduced as d d x c i n ( x ) = c i n ( x + p ) \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 p is a sufficient condition for proving that c i n ( x ) cin\left (x\right ) also satisfies the one-dimensional oscillator equation d 2 d x 2 c i n ( x ) = − c i n ( x ) \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 c i n ( x ) 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 c i n e cine function enables the adoption of independent non-circular definitions of the trigonometric functions s i n e sine and c o s i n e cosine . The properties of c i n e 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.