## Contour integration

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 $|z|<1$ and $\theta$ real define the Cauchy kernel by

$C(\theta,z):=\frac{1}{2\pi}\ \frac{e^{i\theta}}{e^{i\theta}-z}\quad,$

and define the Cauchy transform of a continuous function $g$ on the unit circle $\partial D$ by

$(Cg)(z):=\int_0^{2\pi}g(e^{i\theta})\ C(\theta,z)\ d\theta$

for $|z|<1$. In particular Cauchy’s Formula says that if $g$ is the boundary value of a holomorphic function $f$ on $D$, then $Cg=f$.

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