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Cauchy integral and residue theorem [closed]

What is the difference between the two sets of the following Cauchy integral, $$ \begin{split} \int_c t^k \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=4\pi i \zeta^k\\[8pt] \int_c \frac{1}{t^k} \cdot …
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