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<p>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., it coincides with the horizontal axis in the Argand plane. But what if time makes some angle with the real axis of the Argand plane, rather than completely coinciding with it? We are trying to focus on the case when the time complexity will have both real and imaginary components. For instance, if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41870_2025_2678_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\left( n\right) =\ n\log {n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mfenced close=")" open="("> <mi>n</mi> </mfenced> <mo>=</mo> <mspace width="4pt" /> <mi>n</mi> <mo>log</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, the existing asymptotic notations are capable of handling that in real time. But, if we come across a problem where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41870_2025_2678_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\left( n\right) =\ n\log {n}+i\cdot n^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mfenced close=")" open="("> <mi>n</mi> </mfenced> <mo>=</mo> <mspace width="4pt" /> <mi>n</mi> <mo>log</mo> <mi>n</mi> <mo>+</mo> <mi>i</mi> <mo>·</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41870_2025_2678_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(i=\root 2 \of {-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mroot> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </mroot> </mrow> </math></EquationSource> </InlineEquation>, the existing asymptotic notations will not be able to catch up. To address this issue, we propose the Zeta Notation (<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41870_2025_2678_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\zeta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation>) in our research. This notation qualifies time on both the real and imaginary axes of the Argand plane.</p>

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The Zeta (\(\zeta \)) notation for complex asymptotes

  • 期刊论文

  • 发表日期：

    2025年8月7日

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  • Anurag Dutta

摘要

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., it coincides with the horizontal axis in the Argand plane. But what if time makes some angle with the real axis of the Argand plane, rather than completely coinciding with it? We are trying to focus on the case when the time complexity will have both real and imaginary components. For instance, if \(T\left( n\right) =\ n\log {n}\) T n = n log n , the existing asymptotic notations are capable of handling that in real time. But, if we come across a problem where \(T\left( n\right) =\ n\log {n}+i\cdot n^2\) T n = n log n + i · n 2 , where \(i=\root 2 \of {-1}\) i = - 1 2 , the existing asymptotic notations will not be able to catch up. To address this issue, we propose the Zeta Notation ( \(\zeta \) ζ ) in our research. This notation qualifies time on both the real and imaginary axes of the Argand plane.