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Mathematical methods I course of the University of Oxford

Past exam of the mathematical methods I course of the University of Oxford

2017

For $\delta\notin2\pi\mathbb Z$, the finite <geometric series> gives
$$
\sum_{n=1}^N e^{in\delta}
=\frac{e^{i\delta}(1-e^{iN\delta})}{1-e^{i\delta}}
=e^{i(N+1)\delta/2}\frac{\sin(N\delta/2)}{\sin(\delta/2)}.
$$
The exponential prefactor has <modulus> one. Thus
$$
\boxed{\left|\sum_{n=1}^N e^{in\delta}\right|
=\begin{cases}|\sin(N\delta/2)/\sin(\delta/2)|,&\delta\notin2\pi\mathbb Z,\\N,&\delta\in2\pi\mathbb Z.\end{cases}}
$$
At an excluded phase every summand is one, which supplies the removable limiting value directly. Here $N$ is a positive <integer>.

Solution

Past exam of the mathematical methods I course of the University of Oxford / 2017 / 1 / b / Solution

Past exam of the mathematical methods I course of the University of Oxford / 2017 / 1 / b / Solution

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