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In the text Stein's Complex Analysis the Cauchy Integral formula is given as follows:

$$1.) \, \, \, \, \, \, f(z)=\frac{1}{2\pi i }\int_{C}\frac{f(\zeta)}{\zeta-z}d\zeta$$

How would one appoarch the differentiation of the following identity lead to other integral formulas, essentially any hints or insights into how the Cauchy Integral Formula relates with other integral formulas ?

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    What do you mean ? The most important consequence is that holomorphic $\implies$ analytic https://en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions#Proof expanding $\frac{1}{\zeta-z}$ as a geometric series. And the correct formula is $f(z)=\frac{1}{2i\pi }\int_{C}\frac{f(\zeta)}{\zeta-z}d\zeta$ whenever $f$ is holomorphic inside $C$ – 2017-01-31
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    What I'm essentially asking is how the Cauchy Integral Formula by differentiating it can lead to other integral formulas. – 2017-01-31
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    Suppress the second "=" sign – 2017-01-31

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