Proof
If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proved as a direct consequence of Green's theorem and the fact that the real and imaginary parts of must satisfy the Cauchy–Riemann equations in the region bounded by, and moreover in the open neighborhood U of this region. Cauchy provided this proof, but it was later proved by Goursat without requiring techniques from vector calculus, or the continuity of partial derivatives.
We can break the integrand, as well as the differential into their real and imaginary components:
In this case we have
By Green's theorem, we may then replace the integrals around the closed contour with an area integral throughout the domain that is enclosed by as follows:
However, being the real and imaginary parts of a function analytic in the domain, and must satisfy the Cauchy–Riemann equations there:
We therefore find that both integrands (and hence their integrals) are zero
This gives the desired result
Read more about this topic: Cauchy's Integral Theorem
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