Theoretical and Natural Science

Theoretical and Natural Science

TNS Vol.1 No.1, 05 May 2022

Open Access | Article

Some Fundamental Results from Complex Analysis

Xuhan Pan * 1
1 Beijing Aidi School, Beijing, China

* Author to whom correspondence should be addressed.

Theoretical and Natural Science, TNS Vol.1 No.1, 10-19
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 Xuhan Pan. Some Fundamental Results from Complex Analysis. TNS (2022) TNS Vol.1 No.1: 10-19. DOI: 10.54254/tns.2022003.


This paper is going to introduce the most basic theory of analytic functions of one complex variable. It begins at the discussion of meaning of complex number and the historical development from the formula of cubic equation to the square root of negative number. In the middle section, which is divided in to four small parts. First part states the expression of complex number z, algebraic properties, and the relationship of each single complex number with whole complex plane. Second part concerns about several elementary functions of complex number. Next part relates to the derivative of complex number, such as the partial derivative and Cauchy-Riemann Equation to verdict that whether a function is analytic in the domain. Last part is the integral of complex functions, which concludes some important theorems with the proof, and examples. After the section of introduction of complex number, it comes to the final section which is talking about an identity involving complex number and the prove of it.


Complex Analysis, Complex Variable


1. Ahlfors, L. (1966) Complex analysis. MGH

2. Berlinghoff, W. P. and Gouvea, F. Q. (2015) Math through the ages Vol32. Farmington, Oxton House Publish.

3. Brown, J. W. and Churchill, R. V. (2009) Complex Variables and Applications. McGraw-Hill.

4. Calderon, A. P. (1958) Uniqueness in the Cauchy problem for partial differential equations. The Johns Hopkins University Press.

5. Dyn’kin, E.M. (1972) Investigations in Linear Operators and Function Theory, 128-131.

6. Elias M. Stein. (2003) Complex analysis Vol.2. Princeton University Press, 1-37.

7. Fisher, S. D. (1999) Complex Variables. Dover Publication, 171-183.

8. Gelbaum, R.B. (1992) Problems in real and complex analysis. Springer-Verlag.

9. Hahn, L.S. and Epstein, B. (1996) Classical complex analysis. Jones and Bartlett Publishers

10. Moore, E. H. (1900) Transactions of the American Mathematical Society. American Mathematics Society, 499-506

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 journal agree to the following terms:

1. Authors retain copyright and grant the journal 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 journal.

2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal'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 journal.

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).

Copyright © 2023 EWA Publishing. Unless Otherwise Stated