= Fourier series of a periodic rectangular pulse
{c}
{title2=$a_n=(2/T)\operatorname{sinc}(\pi n w/T)$}
An even train of <rectangular pulses> of width $w<T$, <period> $T$ and unit area per pulse has <Fourier coefficients> $a_n=(2/T)\operatorname{sinc}(\pi nw/T)$ for $n\geq0$. Here the <sinc function> is normalized as $\operatorname{sinc}z=\sin z/z$, and the constant contribution is $a_0/2=1/T$. Integrating the <cosine> over the pulse proves the formula; evenness eliminates the <sine> coefficients. At a pulse edge, <pointwise convergence of a piecewise smooth Fourier series> gives the mean of the one-sided values rather than a prescribed endpoint value.
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