On the Cauchy-Goursat theorem

Abstract: In this study, we have presented a simple and un-conventional proof of a basic but important Cauchy-Goursat theorem of complex integral calculus. The pivotal idea is to sub-divide the region bounded by the simple closed curve by infinitely large number of different simple homotopically clo...

Full description

Bibliographic Details
Main Authors:	Azram, Mohammad, Daoud, Jamal Ibrahim, Elfaki, Faiz Ahmed Mohamed
Format:	Article
Language:	English
Published:	Asian Network for Scientific Information 2010
Subjects:	QA Mathematics
Online Access:	http://irep.iium.edu.my/1769/ http://irep.iium.edu.my/1769/ http://irep.iium.edu.my/1769/ http://irep.iium.edu.my/1769/1/2.htm

id	iium-1769
recordtype	eprints
spelling	iium-17692011-11-21T08:23:01Z http://irep.iium.edu.my/1769/ On the Cauchy-Goursat theorem Azram, Mohammad Daoud, Jamal Ibrahim Elfaki, Faiz Ahmed Mohamed QA Mathematics Abstract: In this study, we have presented a simple and un-conventional proof of a basic but important Cauchy-Goursat theorem of complex integral calculus. The pivotal idea is to sub-divide the region bounded by the simple closed curve by infinitely large number of different simple homotopically closed curves between two fixed points on the boundary. Beauty of the method is that one can easily see the significant roll of singularities and analyticity requirements. We suspect that our approach can be utilized to derive simpler proof for Green ’s theorem, Stoke ’s theorem, generalization to Gauss ’s divergence theorem, extension of Cauchy-Goursat theorem to multiply connected regions, critical study of the affects of singularities over a general field with a general domain and a simpler approach for complex integration such as Cauchy integral formula, residue theorem etc. Avoiding topological and rigor mathematical requirements, we have sub-divided the region bounded by the simple closed curve by a large number of different simple closed curves between two fixed points on the boundary and have introduced: where path of integration is from p to q along ci and for I = 0, 1, 2, ..., n. Line integral along the boundary of the domain was evaluated via ( ∂f/ ∂z). Commutation between integration, δ-operation and d-operation was established. Using the vector interpretation of complex number, the area ds of a small parallelogram was established as . Finally, using Cauchy-Riemann equations we have established the well celebrated Cauchy-Goursat theorem, i.e., if a function f(z) is analytic inside and on a simple closed curve c then Asian Network for Scientific Information 2010 Article PeerReviewed application/pdf en http://irep.iium.edu.my/1769/1/2.htm Azram, Mohammad and Daoud, Jamal Ibrahim and Elfaki, Faiz Ahmed Mohamed (2010) On the Cauchy-Goursat theorem. Journal of Applied Sciences, 10 (13). pp. 1349-1351. ISSN 1812-5662 (O), 1812-5654 (P) http://scialert.net/fulltext/?doi=jas.2010.1349.1351&org=11 DOI: 10.3923/jas.2010.1349.1351
repository_type	Digital Repository
institution_category	Local University
institution	International Islamic University Malaysia
building	IIUM Repository
collection	Online Access
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Azram, Mohammad Daoud, Jamal Ibrahim Elfaki, Faiz Ahmed Mohamed On the Cauchy-Goursat theorem
description	Abstract: In this study, we have presented a simple and un-conventional proof of a basic but important Cauchy-Goursat theorem of complex integral calculus. The pivotal idea is to sub-divide the region bounded by the simple closed curve by infinitely large number of different simple homotopically closed curves between two fixed points on the boundary. Beauty of the method is that one can easily see the significant roll of singularities and analyticity requirements. We suspect that our approach can be utilized to derive simpler proof for Green ’s theorem, Stoke ’s theorem, generalization to Gauss ’s divergence theorem, extension of Cauchy-Goursat theorem to multiply connected regions, critical study of the affects of singularities over a general field with a general domain and a simpler approach for complex integration such as Cauchy integral formula, residue theorem etc. Avoiding topological and rigor mathematical requirements, we have sub-divided the region bounded by the simple closed curve by a large number of different simple closed curves between two fixed points on the boundary and have introduced: where path of integration is from p to q along ci and for I = 0, 1, 2, ..., n. Line integral along the boundary of the domain was evaluated via ( ∂f/ ∂z). Commutation between integration, δ-operation and d-operation was established. Using the vector interpretation of complex number, the area ds of a small parallelogram was established as . Finally, using Cauchy-Riemann equations we have established the well celebrated Cauchy-Goursat theorem, i.e., if a function f(z) is analytic inside and on a simple closed curve c then
format	Article
author	Azram, Mohammad Daoud, Jamal Ibrahim Elfaki, Faiz Ahmed Mohamed
author_facet	Azram, Mohammad Daoud, Jamal Ibrahim Elfaki, Faiz Ahmed Mohamed
author_sort	Azram, Mohammad
title	On the Cauchy-Goursat theorem
title_short	On the Cauchy-Goursat theorem
title_full	On the Cauchy-Goursat theorem
title_fullStr	On the Cauchy-Goursat theorem
title_full_unstemmed	On the Cauchy-Goursat theorem
title_sort	on the cauchy-goursat theorem
publisher	Asian Network for Scientific Information
publishDate	2010
url	http://irep.iium.edu.my/1769/ http://irep.iium.edu.my/1769/ http://irep.iium.edu.my/1769/ http://irep.iium.edu.my/1769/1/2.htm
first_indexed	2023-09-18T20:09:16Z
last_indexed	2023-09-18T20:09:16Z
_version_	1777407367663583232

On the Cauchy-Goursat theorem

Similar Items