- #1

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and got stucked on the 3rd relation.

I don't understand why the Cauchy integral theorem requires that the first term vanishes.

From the Cauchy integral theorem, the contour integral along any path not enclosing a pole is 0. But in this case, the contour [tex]\gamma[/tex] encloses [tex]z_{0}[/tex] which is a pole of [tex](z-z_{0})^{n}[/tex] for [tex]n \in \{-\infty,...,-2\}[/tex]