Finite-part normal trace for mixed quarter-power boundary data ($u=r^{-1/4}(\cos (\theta /4)+\sin (\theta /4))$)

In the upper half-plane, the displayed harmonic function has Dirichlet boundary condition value $x^{-1/4}$ on the positive axis, zero Neumann boundary condition value on the negative axis, and decay at infinity. Its negative trace is $\sqrt{2}|x{|}^{-1/4}$, but its positive normal derivative is $x^{-5/4}/4$. The latter requires a Hadamard finite-part integral in a Fourier transform. For the $e^{ikx}$ convention the transformed traces are $\sqrt{2}\Gamma (3/4)(ik{)}^{-3/4}$ below and $-\Gamma (3/4)(-ik{)}^{1/4}$ above. The homogeneous corner mode $r^{-1/2}\sin (\theta /2)$ produces the additional lower integral transform proportional to $k^{-1/2}$. Higher singular corner modes require a larger distribution class than a locally integrable negative trace.