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We have shown that an extended version of the Cauchy-Goursat theorem holds for all complex analytic functions, anywhere on the complex plane. The extension of the theorem establishes that the integral of any such function is zero, on any closed integration contour within which it is analytic, even it the function has a finite set of isolated singularities on the contour itself, so long as these are all integrable singularities. It is interesting to note that this integrability requirement on the singularities is the minimum necessary requirement for the integral over the contour to make any sense at all, and it is very curious that it turns out to be sufficient for the extension of the Cauchy-Goursat theorem.
The generalization of the result to infinite integration contours with a countable infinity of isolated integrable singularities on them is quite immediate, by a straightforward process of finite induction, using contour manipulation, the extended Cauchy-Goursat theorem and the Cauchy-Goursat theorem in its usual form. Therefore this extended Cauchy-Goursat theorem can be used in essentially all circumstances in which the original Cauchy-Goursat theorem applies, thus giving rise to extensions of many previously known results.
Note that the establishment of this extended Cauchy-Goursat theorem gives
rise immediately to a corresponding extended set of Cauchy integral
formulas, when the interior of the integration contours contain isolated
poles. In fact, these extended Cauchy integral formulas are at the root of
the representation of integrable real functions by inner analytic
functions within the unit disk, which was established in [#!CAoRFI!#]. In
that paper the validity of these extended Cauchy integral formulas is
reflected by the fact that integrals which give the coefficients of the
Taylor series of the inner analytic functions are not only independent of
the radius 
of the circle over which they are calculated, but
are also continuous from within when one approaches the unit circle from
within the open unit disk, that is, in the
limit.
Since the Cauchy-Goursat theorem is such an important and fundamental one, it is to be expected that this extension will have further interesting consequences, possibly in many fields of mathematics and also in many applications in physics.