Upper-quadrant cancellation of reflected boundary transforms

If $P(k)$ is analytic in the upper half-plane with suitable growth, paired integrals along the positive real and imaginary rays cancel when their exponential is $e^{ik(x+iy)}$ with $x,y>0$. For $0<y<\ell$, the analogous pair along the imaginary and negative real rays cancels with $e^{ik(x+i(y-\ell ))}$. These two applications of the Cauchy integral theorem eliminate reflected unknown transforms from a strip global relation; an individual ray integral need not vanish.