= Restricted Mellin transform on the unit interval
{title2=$H(s)=\int_0^1\lambda^{s-1}h(\lambda)\,d\lambda$}
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 $h(\lambda)=\lambda^q\mathbf1_{(0,c)}(\lambda)$ has transform $c^{s+q}/(s+q)$ when $\operatorname{Re}(s+q)>0$. Multiplication by $c^s$ shifts the support threshold in logarithmic coordinates, the <Laplace transform time-shift rule> expressed multiplicatively.
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