= Box smoothing of a two-sided exponential
{title2=$w(x)=(2a)^{-1}\int_{-a}^a(2b)^{-1}e^{-|x-t|/b}\,dt$}
For $a,b>0$, the <convolution> of the normalized interval indicator $(2a)^{-1}\mathbf1_{[-a,a]}$ with the two-sided exponential $(2b)^{-1}e^{-|x|/b}$ is
$$
w(x)=\begin{cases}[1-e^{-a/b}\cosh(x/b)]/(2a),&|x|\leq a,\\\sinh(a/b)e^{-|x|/b}/(2a),&|x|\geq a.\end{cases}
$$
Split the defining integral at $t=x$ in the interior case; outside the interval the absolute value has a fixed sign. These elementary integrals prove the formula and its continuity at the joining points. With <Fourier transform> convention $\widehat w(k)=\int w(x)e^{-ikx}dx$, the <convolution theorem> gives $\widehat w(k)=\sin(ak)/[ak(1+b^2k^2)]$, using its continuous value at $k=0$. <Parseval's identity> can therefore turn the squared integral of this rational-trigonometric transform into a piecewise elementary real-space integral.
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