= Finite-horizon extension of a boundary transform
{title2=$G(k,t)\longmapsto G(k,T),\quad T>t$}
In a <Fokas method> boundary <integral> over $\partial D_+$, where $\operatorname{Re}\omega<0$ inside, the contribution of replacing $G(k,t)$ by $G(k,T)$ is an <integral> of $e^{ikx}\int_t^T e^{\omega(k)(s-t)}g(s)\,ds$ times an <holomorphic> coefficient. For $x>0$ it vanishes by the <Cauchy integral theorem> and the exponential decay in the upper domain. The transformed amplitude is then independent of the observation time, which simplifies differentiation of the <partial differential equation>. Temporal inversion at $0<t<T$ recovers the full boundary value rather than the half-value of a truncated transform at its endpoint.
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