The Zeta ($ζ$) Notation for Complex Asymptotes

TL;DR

The paper addresses extending asymptotic notation to encompass imaginary components of time complexity by introducing the Zeta notation on the Argand plane. It defines $T(n)=\zeta(g(n),\phi)=e^{i\phi}g(n)$ with $g(n)$ and $\phi$ derived from $f(n)$, linking the real and imaginary parts via Euler's formula. It demonstrates a one-degree transformation to connect Zeta notation with classical real-domain asymptotics and discusses equivalence conditions when the imaginary part vanishes. The work argues that existing notations are limited to real growth, while Zeta provides a framework to capture and transform imaginary complexities, with future directions in complex-analysis-based refinements.

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 ($ζ$), which would qualify Time in both the Real and Imaginary Axis, as per the Argand Plane.

Figures (6)

• Figure 1: Graphical Implication of the Real valued asymptotic notations. The ordinates here stand for the computational time, while the abscissae implicate the cardinally of the data input.
• Figure 2: An example of a complex plane. Upon this vertical spatial alliance are just the imaginary numbers. On the Horizontal spatial alliance, are the real numbers. The angle $\phi$ is the one that the Complex Number ($z$) subtends on the Real Axes. According to this illustration, multiplying by -1 causes a half circle's origin to rotate $180^o$. A spin of $90^o$ around the center, which corresponds to a quarter of a circle, results from multiplying by $i$.
• Figure 3: Graphical Representation of the Computational Complexity with consideration of the Imaginary Part along with the cardinality of the Dataset.
• Figure 4: Graphical Representation of the Computational Complexity before being spitted in terms of its trigonometric components in a bi-dimensional plane.
• Figure 5: Pictorial Representation of the $\zeta$ Notation on the Polar Coordinate System. The circle is having a radial length of $\sqrt[2]{\left(\text{Re}\left(f(n)\right)\right)^2+\left(\text{Im}\left(f(n)\right)\right)^2}$ and will subtend an angle of ${\text{tan}}^{-1}{\left(\frac{\text{Im}\left(f(n)\right)}{\text{Re}\left(f(n)\right)}\right)}$ with the positive $x$ axes.