Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 86 / 3 / b / Solution

Since $q$ is harmonic, $q_z$ is holomorphic and the one-form $e^{-ikz}{q}_{z}dz$ is closed. Closing the physical quadrant at infinity gives the global relationIn this third spectral quadrant the exponential decays on both boundary axes. Set $\gamma ={\beta }_1+{\beta }_2$. The relation becomes

All four boundary functions are real. Consequently their transforms obeyand the same identities for $J_1,J_2$. Complex conjugation of the global relation therefore gives its upper-left consequenceIntroduce just one unknown function,It is holomorphic in the upper half-plane. Under the transform-admissible decay assumptions, $|\Phi (k)|\le {\int }_0^{\infty }(|h_2|+|j_2|)dx$, so it is bounded there.

For $k$ on the positive real ray, use both consequences of the global relation at $-k$. Adding them removes $J_1(-k)$ and givesFor $k$ on the positive imaginary ray, use the upper-left relation and $H_2(-k)+i{J}_2(-k)=2H_2(-k)-\Phi (k)$. The resulting single-function spectral elimination for a rational-angle quadrant isBoundary limits from the appropriate transform domains are understood. Every remaining unknown boundary derivative is contained in the one bounded upper-half-plane function $\Phi$.