Theoretical and Natural Science

Theoretical and Natural Science

TNS Vol.1 No.1, 05 May 2022

Open Access | Article

Some Fundamentals of Complex Analysis

Kewei Huang * 1 , Zijie Shen 2 , Tianyi Zhang 3
1 Dulwich International High School Suzhou, Suzhou, 215000, China
2 Guangzhou Panyu Clifford International School, Guangzhou, 510000, China
3 Shanghai Pinghe School, Shanghai, 201206, China

* Author to whom correspondence should be addressed.

Theoretical and Natural Science, TNS Vol.1 No.1, 51-64
Published 05 May 2022. © 2023 The Author(s). Published by EWA Publishing
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.
Citation Kewei Huang, Zijie Shen, Tianyi Zhang. Some Fundamentals of Complex Analysis. TNS (2022) TNS Vol.1 No.1: 51-64. DOI: 10.54254/tns.2022008.


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.


Complex Analysis, Complex Variables


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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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