Theoretical and Natural Science
TNS Vol.1 No.1, 05 May 2022
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Some Fundamentals of Complex Analysis
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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.
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