 Solution

Fourier transformation givesFor $x>0$, close the inverse-transform contour in the upper half-plane and take the residue at $k=im$; for $x<0$, close it in the lower half-plane. This gives the one-dimensional modified Helmholtz Green functionThe same formula works for complex $m$ with $\mathrm{Re}m>0$: it decays at both ends, is continuous at zero, and its derivative has the jump $G^{\prime }(0+)-G^{\prime }(0-)=-1$ required by the delta source.

Convolution therefore yields