1. Global nature

An important feature to note about Cauchy’s theorem is the *global* nature of its hypothesis on the analytic functions f. Cauchy’s theorem is the powerful technique of *contour shifting*, which allows one to compute a contour integral by replacing the contour with a homotopic contour on which the integral is easier to either compute or integrate.

2. Rigidity for non-analytic

3. Analyticity criterion: Morera’s theorem

4. Cauchy’s theorem, Poisson Kernel and Brownian motion

For and $\theta$ real define the **Cauchy kernel** by

and define the **Cauchy transform** of a continuous function on the unit circle by

for . In particular Cauchy’s Formula says that if is the boundary value of a **holomorphic** function on , then .

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