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Metric spaces and complex analysis course of the University of Oxford

Past exam of the metric spaces and complex analysis course of the University of Oxford

2017

For $\phi(t)=re^{it}$, $\phi'(t)=ire^{it}$ and $|\phi'(t)|=r$. The <triangle inequality for an integral> applied to the parametrized <contour integral> gives
$$
\boxed{\left|\int_\phi f(z)\,dz\right|=\left|\int_0^{2\pi}f(re^{it})ire^{it}\,dt\right|\leq r\int_0^{2\pi}|f(re^{it})|\,dt.}
$$
This is the circular-contour instance of the <estimation lemma>.

Solution

Past exam of the metric spaces and complex analysis course of the University of Oxford / 2017 / 3 / a / ii / Solution

Past exam of the metric spaces and complex analysis course of the University of Oxford / 2017 / 3 / a / ii / Solution

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Past exam of the metric spaces and complex analysis course of the University of Oxford / 2017 / 3 / a / ii / Solution

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