Franklin's Notes


Cauchy integral formula

The Cauchy integral formula is as follows:

Cauchy integral formula. If $f(z)$ is holomorphic in the open disk $\Delta$, and $\gamma$ is a closed curve in $\Delta$, then for any point $a\in\Delta\backslash\gamma$, we have where $\mathrm{ind}(\gamma,a)$ is the winding number .

One way of proving this fact is to consider the function which is guaranteed to be holomorphic everywhere in $\Delta$ except at $z=a$. However, even at this exceptional point, we have that Hence, we may apply Cauchy's Theorem for disks to conclude that or equivalently which may be rearranged to obtain the equation Of course, the latter integral is equal to $2\pi i$ times the index of $\gamma$ about $a$, so we have the formula as desired.

complex-analysis

complex-integration

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