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Past exam of the mathematics course of the University of Oxford

2017

Prelims

Mathematics V

{c}

The real <Fourier series> of $f$ is written
$$
\boxed{f(x)\sim\frac{a_0}{2}+\sum_{n=1}^{\infty}\bigl(a_n\cos nx+b_n\sin nx\bigr),}
$$
where the <Fourier coefficients> are
$$
\boxed{
\begin{aligned}
a_0&=\frac1\pi\int_{-\pi}^{\pi}f(x)\,dx,\\
a_n&=\frac1\pi\int_{-\pi}^{\pi}f(x)\cos nx\,dx,\qquad n\geq1,\\
b_n&=\frac1\pi\int_{-\pi}^{\pi}f(x)\sin nx\,dx,\qquad n\geq1.
\end{aligned}}
$$
By part (a)(i), any other interval of length $2\pi$ gives the same <Fourier coefficients>, because each integrand is a <periodic function> with period $2\pi$. The notation $\sim$ associates $f$ with its <Fourier series>; existence of the <integrals> alone does not assert <pointwise convergence>.

Solution

Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V / 4 / b / Solution

Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V / 4 / b / Solution

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Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V / 4 / b / Solution

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