Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 13E / b / Solution

Use the principal square root and deform the Bromwich contour around the negative real-axis branch cut. The pole at $p=0$ contributes $a$. The two boundary values at $p=-b$ combine toOn the cut put $p=-v^2$; the jump of $e^{-x\sqrt{p/\kappa }}$ is $-2i\sin (xv/\sqrt{\kappa })$ and $dp=-2v,dv$. Taking the principal value at $v=\sqrt{b}$ givesThe expression has the prescribed boundary value and decays into the bar.