INTRODUCTION
Complex variables open everything what is hidden in the real calculus. Complex integration is central in the study of complex variables. As in calculus, the fundamental theorem of calculus is significant because it relates integration with differentiation and at the same time provides method of evaluating integral so is the complex analog to develop integration along arcs and contours is complex integration. Complex integration is elegant, powerful and a useful tool for mathematicians, physicists and engineers. CauchyGoursat theorem is a fundamental, well celebrated theorem of the complex integral calculus. This theorem is not only a pivotal result in complex integral calculus but is frequently applied in quantum mechanics, electrical engineering, conformal mappings, method of stationary phase, mathematical physics and many other areas of mathematical sciences and engineering. It provides a convenient tool for evaluation of a wide variety of complex integration. It also forms the cornerstone of the development of results f'(z) of an analytic function f(z) is analytic, Cauchy ’s integral formula and many advance topics in complex integration.
Due to its pivotal role and importance, mathematicians have discussed it in
all respects (Gario, 1981; Gurtin
and Martins, 1976; Mibu, 1959; Segev
and Rodnay, 1999).
Historically, it was firstly established by Cauchy in 17891857 and Churchill
and James (2003) and later on extended by Goursat in 18581936 and Churchill
and James (2003) without assuming the continuity of f' (z). Consequently,
it has laid down the deeper foundations for Cauchy Riemann theory of complex
variables. Its usual proofs involved many topological concepts related to paths
of integration; consequently, the reader especially the undergraduate students
can not be expected to understand and acquire a proof and enjoy the beauty and
simplicity of it. Hence, it will not be unusual to motivate CauchyGoursat theorem
by a simple version (Gurin, 1981; Long,
1989; Tucsnak, 1984).
In this study, we have adopted a simple nonconventional approach, ignoring
some of the strict and rigor mathematical requirements. Knowledge of calculus
will be sufficient for understanding. For further reading, reader can choose
any standard book on complex variables and/or calculus (Churchill
and James, 2003; Mathews and Russell, 2006). 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 will be useful without any difficulty 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.
RESULTS AND DISCUSSION
Statement of Cauchygoursat theorem: If a function f(z) is analytic
inside and on a simple closed curve c then
.
Proof: Let f(z) = f(x+iy) = f(x, y) = u(x, y)+iv(x, y) be analytic inside
and on a simple closed curve c. Need to prove that .
Consider the region R enclosed by simple closed curve C as R = {(x,y) p ≤x ≤q and g1(x) ≤y ≤g2(x)}. For the sake of proof, assume C is oriented counter clockwise. Let p and q be two fixed points on C (Fig. 1). Subdivide the region enclosed by C, by a large number of paths c_{0}, c_{1}, c_{2},..., c_{n} passing through the points p and q with c_{0} = {(x, y) p ≤x ≤q, y = g_{1} (x)} and c_{0} = {(x, y) p ≤x ≤q, y = g_{2} (x)} constituting the boundary of C (Fig. 1).
Define
and

Fig. 1: 
Region R enclosed by simple closed curve C: 
Now:
Note that for large n, δI_{i} = I_{i}I_{i+1}
is the small variation in the values of the integrals along two adjacent homotopically
close lying paths. Since, f(z) is analytic so we can enjoy commutation between
integration and δoperation as follow:
From physical interpretation, δoperation and doperation on z are commutative, i.e.,
Consequently, integrating by parts the 2nd integral of Eq. 1
and considering the fact that there is no variation in z at the fixed points
p and q, Eq. 1 reduces as:
Hence,
Now considering the function ds as a function of complex conjugate coordinates, i.e.,
Eq.
3 changes as:
Now, using the vector interpretation of complex number, the area ds of a small
parallelogram is given by Consequently,
Eq. 4 reduces as:
This completes the proof
CONCLUSION
CauchyGoursat theorem is the basic pivotal theorem of the complex integral
calculus. The present proof avoids most of the topological as well as strict
and rigor mathematical requirements. Instead, standard calculus results are
used. Line integral of f(z) around the boundary of the domain, e.g., along C
has been evaluated via ( ∂f/ ∂z). In general ( ∂f/ ∂z) may
not be zero but analyticity of f(z) 0plays a pivotal roll for ( ∂f/ ∂z).
It is also interesting to note the affect of singularities in the process of
subdivision of the region and line integrals along the boundary of the regions.
I suspect this approach can be considered over any general field with any general
domain.