The Zeta ($\zeta$) Notation for Complex Asymptotes

Abstract

Time Complexity is an important metric to compare algorithms based on their cardinality. The commonly used, trivial notations to qualify the same are the Big-Oh, Big-Omega, Big-Theta, Small-Oh, and Small-Omega Notations. All of them, consider time a part of the real entity, i.e., Time coincides with the horizontal axis in the argand plane. But what if the Time rather than completely coinciding with the real axis of the argand plane, makes some angle with it? We are trying to focus on the case when the Time Complexity will have both real and imaginary components. For Instance, if $T\left(n\right)=\ n\log{n}$, the existing asymptomatic notations are capable of handling that in real time But, if we come across a problem where, $T\left(n\right)=\ n\log{n}+i\cdot n^2$, where, $i=\sqrt[2]{-1}$, the existing asymptomatic notations will not be able to catch up. To mitigate the same, in this research, we would consider proposing the Zeta Notation ($\zeta$), which would qualify Time in both the Real and Imaginary Axis, as per the Argand Plane.