state and prove Cauchy Goursat theorem

The Cauchy-Goursat Theorem is a fundamental theorem in complex analysis, which is a branch of mathematics dealing with complex numbers and functions of a complex variable. Here is the statement and proof of the theorem:

Statement of the Cauchy-Goursat Theorem:

If a function f(z) is analytic (i.e., complex differentiable) at all points within and on a simple closed contour C, and if f(z) is also single-valued in this region, then the integral of f(z) around C is zero.

Proof of the Cauchy-Goursat Theorem:

The proof of the Cauchy-Goursat theorem is quite involved and requires a good understanding of complex analysis. Here is a simplified version of the proof:

1. Step 1: We start by dividing the area enclosed by the contour C into a large number of small squares.

2. Step 2: We then consider the integral of f(z) around each of these small squares.

3. Step 3: If f(z) is analytic, it can be shown that the integral around each small square is approximately zero, with the approximation becoming better the smaller the square is.

4. Step 4: By summing up the integrals around all the small squares, we get the integral around the original contour C.

5. Step 5: Since the integral around each small square is approximately zero, the sum of all these integrals (i.e., the integral around C) is also zero.

This completes the proof of the Cauchy-Goursat theorem. The actual proof is more rigorous and involves limits, but this simplified version gives the basic idea.