A Novel Step Function for Operational Calculus

This is a novel Unit-Step function that was developed while working with Fourier Transforms and the Heavyside Step Function in an attempt to remove discontinuities at boundaries of piecewise functions and formulate waves that are primally periodic. The function's main utility is to offer a method to construct single function alternatives to plurally defined piecewise functions. The form is interesting in relation to its components considering that pi, i, and the natural log function usually invoke imagery of curves, complexity, and irrationality compared to the resulting relatively simplistic symbolic symmetry and graphically flat binary step function. At some future point I may, or as others are inclined, explore the functions implications through related topics of operational calculus. The panels are as follows:

Panel 1 is is the full function, showing the relation between k and x where k is the location on the x-axis of the step.

Panel 2 - An example with k=5

Panel 3 - The limit at 0.

Panel 4 & 5 - A symmetrical version and example.

Panel 6 through 9 - Compositions made from symmetry variants showing a valley, a plateau, stairs, and a chain.