= Boundary spectral representation of a holomorphic function on a semistrip
{title2=$f(z)=\frac1{2\pi}\left(\int_0^{i\infty}e^{ikz}\rho_L\,dk+\int_0^\infty e^{ikz}\rho_B\,dk+\int_0^{-\infty}e^{ikz}\rho_T\,dk\right)$}
For a <holomorphic function> on $x>0$, $0<y<l$, define boundary spectra by counterclockwise traversal: $\rho_L=\int_{il}^0e^{-ikw}f(w)\,dw$, $\rho_B=\int_0^\infty e^{-ikx}f(x)\,dx$, and $\rho_T=-e^{kl}\int_0^\infty e^{-ikx}f(x+il)\,dx$. The displayed reconstruction follows from the <Cauchy integral formula> and the ray identity $\int_0^{e^{i\theta}\infty}e^{ik(z-w)}\,dk=i/(z-w)$ whenever that exponential decays. The three rays are selected by the inward normals of the three sides.
// Target: analysis.bigb
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