Restricted Mellin transform on the unit interval ($H(s)={\int }_0^1{\lambda }^{s-1}h(\lambda )d\lambda$)

This is the Mellin transform of the function extended by zero beyond the unit interval. The change of variables $\lambda =e^{-t}$ identifies it with the unilateral Laplace transform of $h(e^{-t})$. The Bromwich inversion formula therefore gives $h(\lambda )=(2\pi i{)}^{-1}{\int }_{\gamma -i\infty }^{\gamma +i\infty }{\lambda }^{-s}H(s)ds$ for $0<\lambda <1$, with $\gamma$ in the convergence half-plane and suitable inversion hypotheses. At a jump, symmetric inversion returns the mean of the two limiting values. In particular, for $0<c<1$, the cutoff $h(\lambda )={\lambda }^q{1}_{(0,c)}(\lambda )$ has transform $c^{s+q}/(s+q)$ when $\mathrm{Re}(s+q)>0$. Multiplication by $c^s$ shifts the support threshold in logarithmic coordinates, the Laplace transform time-shift rule expressed multiplicatively.