Boundary spectral representation of a holomorphic function on a semistrip ($f(z)=\frac{1}{2\pi }\left({\int }_0^{i\infty }{e}^{ikz}{\rho }_{L}dk+{\int }_0^{\infty }{e}^{ikz}{\rho }_{B}dk+{\int }_0^{-\infty }{e}^{ikz}{\rho }_{T}dk\right)$)

For a holomorphic function on $x>0$, $0<y<l$, assume its boundary traces give convergent displayed integrals and that its far-end contour contribution tends to zero. Define boundary spectra by counterclockwise traversal: ${\rho }_L={\int }_{il}^0{e}^{-ikw}f(w)dw$, ${\rho }_B={\int }_0^{\infty }{e}^{-ikx}f(x)dx$, 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