= Finite-part normal trace for mixed quarter-power boundary data
{title2=$u=r^{-1/4}(\cos(\theta/4)+\sin(\theta/4))$}
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 $\sqrt2|x|^{-1/4}$, 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 $\sqrt2\Gamma(3/4)(ik)^{-3/4}$ 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 (mathematical analysis)> class than a locally integrable negative trace.
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