Azram, Mohammad and Daoud, Jamal Ibrahim and Elfaki, Faiz Ahmed Mohamed (2010) On the CauchyGoursat theorem. Journal of Applied Sciences, 10 (13). pp. 13491351. ISSN 18125654 EISSN 18125662
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Abstract
Abstract: In this study, we have presented a simple and unconventional proof of a basic but important CauchyGoursat theorem of complex integral calculus. The pivotal idea is to subdivide 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 CauchyGoursat 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 subdivided 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, δoperation and doperation was established. Using the vector interpretation of complex number, the area ds of a small parallelogram was established as . Finally, using CauchyRiemann equations we have established the well celebrated CauchyGoursat theorem, i.e., if a function f(z) is analytic inside and on a simple closed curve c then
Item Type:  Article (Journal) 

Additional Information:  3127/1769 
Uncontrolled Keywords:  CauchyGoursat theorem, analytic function, simple closed curve, singular points, CauchyRiemann equations 
Subjects:  Q Science > QA Mathematics 
Kulliyyahs/Centres/Divisions/Institutes (Can select more than one option. Press CONTROL button):  Kulliyyah of Engineering > Department of Science 
Depositing User:  Prof Mohammad Azram 
Date Deposited:  09 Sep 2011 09:04 
Last Modified:  01 Aug 2022 10:20 
URI:  http://irep.iium.edu.my/id/eprint/1769 
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