Exercise · ZW-CAUCHY-GOURSAT-THEOREM-007 · ★★☆ Intermediate

Let \(D\) be simply connected and \(\oint_C f\,dz\neq 0\) for some closed contour \(C\) in \(D\). What follows?

Choose one option.

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  • A. \(f\) is not holomorphic at some point of \(D\)

Explanation

This is the contrapositive of the theorem: if \(f\) were holomorphic on \(D\), every closed integral would be \(0\). A primitive would also force it to be \(0\).

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Review the concept: Cauchy-Goursat theorem

The Cauchy-Goursat theorem says that, if \(f\) is holomorphic on a simply connected open set \(D\), its integral over any closed contour \(C\) contained in \(D\) is zero.

\[\oint_C f(z)\,dz=0\]

More practice problems: cauchy-goursat theorem

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