Let C be the circle |z|=2 traversed once. Use |∫_C f dz|≤ ML with f(z)=1/z^2 to bound |∫_C f dz|.

Exercise · ZW-CONTOUR-INTEGRAL-009 · ★★★ Advanced

Let \(C\) be the circle \(|z|=2\) traversed once. Use \(\left|\int_C f\,dz\right|\le ML\) with \(f(z)=1/z^2\) to bound \(\left|\int_C f\,dz\right|\).

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B. \(\le\pi\)

Explanation

On \(|z|=2\): \(|f|=1/4\), so \(M=\tfrac14\), and the length is \(L=4\pi\). Then \(ML=\pi\). (In fact the integral is \(0\), but the bound only guarantees \(\le\pi\).)

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Review the concept: Contour integral

The contour integral of a continuous function \(f\) over a contour \(C\) with parametrization \(\gamma\) is the limit of sums of \(f\) evaluated at points of \(C\) times the small complex displacements.

\[\int_C f(z)\,dz=\int_a^b f(\gamma(t))\,\gamma'(t)\,dt\]

Let \(C\) be the circle \(|z|=2\) traversed once. Use \(\left|\int_C f\,dz\right|\le ML\) with \(f(z)=1/z^2\) to bound \(\left|\int_C f\,dz\right|\).

Explanation

Review the concept: Contour integral

More practice problems: contour integral