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Mathematics and philosophy course of the University of Oxford

Year 2 of the mathematics and philosophy course of the University of Oxford

Algebra and analysis II for mathematics and philosophy course of the University of Oxford

Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford

2013

<Liouville's theorem> states that every <bounded> <entire function> is constant:
$$
\boxed{f\text{ holomorphic on }\mathbb C,\quad\sup_{z\in\mathbb C}|f(z)|<\infty\quad\Longrightarrow\quad f\text{ constant}.}
$$
For example, the Cauchy <derivative> estimate on a <circle> of radius $R$ gives $|f'(z)|\leq M/R$. Letting $R\to\infty$ gives $f'=0$ everywhere, proving the conclusion.

Solution

Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 6 / c / Solution

Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 6 / c / Solution

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Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 6 / c / Solution

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