Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)
Series Vol. 5 , 25 May 2023
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Complex Analysis and Residue Theorem
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Abstract
The study of the properties of analytical functions is described as complex analysis. The residue theorem is an important conclusion in complex analysis. This paper introduces the origin of imaginary numbers from Cardano Formula defines the conversion of complex number formats from the Euler Theorem, and proves the intermediate theorem Cauchy Integral Formula before reaching our final conclusion and the goal of the paper, Residue Theorem. The name of the theorem comes from the concept of residue, which is defined using a function’s Laurent series. We could then derive the Residue Theorem from the Cauchy Integral Theorem, also called the Cauchy-Goursat Theorem. We will be able to formalize our prior, ad hoc method of computing integrals over contours encompassing singularities. Additionally, it is a theorem that may be applied to zero-pole qualities and curved integral properties. The Residue Theorem is the basis of many essential mathematical facts revolving around line integrals, particularly in solving ODEs and PDEs and describing physics models.
Keywords
residue theorem, complex analysis, residue.
References
1. Michel Brion and Michèle Vergne. “Residue formulae, vector partition functions and lattice points in rational polytopes.” AMS, Accessed 10 September 2022.
2. Levine, Marc, and Sabrina Pauli. “Quadratic Counts of Twisted Cubics - math.” arXiv, 12 June 2022, Accessed 10 September 2022.
3. Beck, Matthias. “Counting Lattice Points by Means of the Residue Theorem - The Ramanujan Journal.” Springer, Accessed 10 September 2022.
4. Bliard, Gabriel. “Notes on $n$-point Witten diagrams in AdS${}_2$.” arXiv, 4 April 2022, Accessed 10 September 2022.
5. Ce Xu, Lu Yan. “Parametric Euler T-sums of odd harmonic numbers.” ArXiv, Submitted on 26 Mar 2022.
6. Humberto Gomez, Renann Lipinski Jusinskas, Arthur Lipstein. “Cosmological Scattering Equations at Tree-level and One-loop.” arXiv, 23 December 2021, Accessed 10 September 2022.
7. Tyler Gorda, Juuso Österman, Saga Säppi. “Augmenting the residue theorem with boundary terms in finite-density calculations.” arXiv, 30 August 2022, Accessed 10 September 2022.
8. Neng Wan, Dapeng Li, Lin Song, Naira Hovakimyan. “Simplified Analysis on Filtering Sensitivity Trade-offs in Continuous- and Discrete-Time Systems.” arXiv, 8 April 2022, Accessed 10 September 2022.
9. B. Ananthanarayan, Sumit Banik, Samuel Friot, Tanay Pathak. “On the Method of Brackets.” arXiv, 17 December 2021, Accessed 10 September 2022.
10. Chamizo, Fernando. “A simple evaluation of a theta value and the Kronecker limit formula.” arXiv, 18 July 2021. Accessed 10 September 2022.
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2023)
- ISBN (Print)
- 978-1-915371-53-9
- ISBN (Online)
- 978-1-915371-54-6
- Published Date
- 25 May 2023
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/5/20230307
- Copyright
- © 2023 The Author(s)
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited