Exercise · ZW-CAUCHY-INTEGRAL-FORMULA-009 · ★★☆ Intermediate

Which hypothesis is needed to apply the Cauchy integral formula to \(\oint_C\frac{f(z)}{z-z_0}dz\)?

Choose one option.

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C. \(f\) holomorphic on \(C\) and inside it, and \(z_0\) inside \(C\)

Explanation

Holomorphy is needed on the whole enclosed region, boundary included, and \(z_0\) must be in the interior. If \(z_0\) is outside, the integral is zero; if it lies on \(C\), the integrand is undefined.

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Review the concept: Cauchy integral formula

The Cauchy integral formula gives the value of a holomorphic function \(f\) at a point \(z_0\) inside a simple closed contour \(C\) by means of an integral of \(f\) over \(C\).

\[f(z_0)=\frac{1}{2\pi i}\oint_C\frac{f(z)}{z-z_0}\,dz\]

Which hypothesis is needed to apply the Cauchy integral formula to \(\oint_C\frac{f(z)}{z-z_0}dz\)?

Explanation

Review the concept: Cauchy integral formula

More practice problems: cauchy integral formula