Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 2, 20 February 2023
Open
Access | Article
Some Fundamentals of Complex Analysis
* Author to whom correspondence should be addressed.
Abstract
In this paper, we first introduced the significance of complex variables as a field in the first section. Then, for the main body section, we explored the history and development of complex numbers. In addition, we demonstrated the key concepts and theorems that are crucial in the fundamentals of this field. We dive in depth to the such fundamental knowledge, which is the preliminaries of complex numbers. We analyzed the preliminaries via three smaller sections: complex numbers and the complex plane, functions on the complex plane, and integration along curves. Finally, we applied these knowledges into deriving a proof for the product of different sine expressions, which results in a generalized formula. Ultimately, the last section will be the conclusion of the paper.
Keywords
Complex Analysis, Complex Variables
References
1. L. V. Ahlfors. Conformal Invariants. McGraw-Hill, New York, 1973.
2. L. V. Ahlfors. Complex Analysis. McGraw-Hill, New York, third edition, 1979.
3. G. B. Airy. On the intensity of light in the neighbourhood of a caustic. Transactions of the Cambridge Philosophical Society, 6:379– 402, 1838.
4. J. Bak and D. J. Newman. Complex Analysis. Springer-Verlag, New York, second edition, 1997.
5. B. Blank, An Imaginary Tale Book Review, in Notices of the AMS Volume 46, Number 10, November 1999, pp. 1233-1236.
6. H. Dym and H. P. McKean: Fourier Series and Integrals, Academic Press, 1972.
7. T. W. Körner: Fourier Analysis, Cambridge University Press, 1988.
8. J. S. Walker: Fourier Analysis, Oxford University Press, 1988.
9. E.T. Whittaker and G.N. Watson. A Course in Modern Analysis. Cambridge University Press, 1927.
10. E.M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, 2003.
11. Taylor’s Theorem and Applications, James S. Cook, November 11, 2018, For Math 132 Online.
12. Taylor & Laurent theorem, Chandan kumar Department of physics S N Sinha College Jehanabad Introduction.
13. A Formal Proof of Cauchy’s Residue Theorem, Wenda Li and Lawrence C. Paulson Computer Laboratory, University of Cambridge.
14. Robert B.Ash Complex Variables.” Academic Press. Ash, Robert B.. “Chapter Title.” Book Title: Subtitle, edited by Editor name, Publisher, Year, pp. Page range. 1971
15. Zagier, Don. Newman's short proof of the prime number theorem. American Mathematical Monthly. 104 (8): 705–708. doi:10.2307/2975232. JSTOR 2975232. MR 1476753. 1997
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. Authors who publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See Open Access Instruction).
- Volume Title
- Proceedings of the International Conference on Computing Innovation and Applied Physics (CONF-CIAP 2022)
- ISBN (Print)
- 978-1-915371-13-3
- ISBN (Online)
- 978-1-915371-14-0
- Published Date
- 20 February 2023
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/2/20220090
- 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