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Geometry course of the University of Oxford

Past exam of the geometry course of the University of Oxford

2007

Choose $r>0$ so the closed disk centered at $a$ of radius $r$ lies in $U$. The <Cauchy integral formula> gives
$$
f(a)=\frac1{2\pi i}\int_{|z-a|=r}\frac{f(z)}{z-a}dz.
$$
Parametrize $z=a+re^{i\theta}$; then $dz=ire^{i\theta}d\theta$ and the denominator cancels. Thus
$$
\boxed{f(a)=\frac1{2\pi}\int_0^{2\pi}f(a+re^{i\theta})d\theta.}
$$

Solution

Past exam of the geometry course of the University of Oxford / 2007 / B / 8 / ii / Solution

Past exam of the geometry course of the University of Oxford / 2007 / B / 8 / ii / Solution

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Past exam of the geometry course of the University of Oxford / 2007 / B / 8 / ii / Solution

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