= Exponential-decay contour for half-line drift diffusion
{title2=$k=i\alpha+r e^{i\theta},\quad k=i\alpha+r e^{i(\pi-\theta)}$}
For $q_t=q_{xx}+\alpha q_x$ with $\alpha>0$, the <dispersion relation> is $\omega=k^2-i\alpha k$ and its symmetry is $\nu=i\alpha-k$. On the upper domain $\operatorname{Re}\omega<0$, $\operatorname{Im}k>\alpha$ and therefore $\operatorname{Im}\nu<0$. After removing the unknown trace through the <global relation>, deform the boundary <contour> to the displayed two rays, with $0<\theta<\pi/4$, oriented from left infinity to $i\alpha$ to right infinity. The factor $e^{ikx}$ decays like $e^{-r x\sin\theta}$, while $e^{-\omega t}$ has Gaussian decay for large $r$. Apparent poles introduced by splitting a finite-time forcing transform must be treated as removable singularities of the unsplit expression.
Back to article page
