Spiral Exponential Function—Chapter 6: The Spiral Exponential Function

This chapter explores the spiral exponential function, a unified framework that encompasses divergent, rotational (oscillatory), and convergent motion modes through a single mathematical structure. Building on the Maclaurin expansion and Euler's formula, it introduces a mediating angle to smoothly transition between real exponential and complex forms, including trigonometric and hyperbolic functions. The spiral exponential function, defined as , reveals a geometric continuity on Euler's circle and connects naturally to the Laplace transform form, offering applicability in fields such as control engineering.