Journal of Applied Sciences1812-56541812-5662Asian Network for Scientific Information10.3923/jas.2010.1349.1351AzramM.I. DaoudJamalA.M. ElfakiFaiz1220101013In 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 c_{i} 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 .]]>Churchill, R.V. and W.B. James,2003Gario, P.,1981Gurin, A.M.,1981Long, W.T.,1989Gurtin, M.E. and L.C. Martins,1976Mathews, J.H. and W.H. Russell,2006Mibu, Y.,1959Segev, R. and G. Rodnay,1999Tucsnak, M.,1984