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Integrated version of the Post-Widder inversion formula for Laplace transforms. (English) Zbl 1236.44003
The formula mentioned in the title reads $f(t)= \lim_{n\to\infty}(-1)^n/(n!\Gamma(\alpha))\int^t_0 (t-s)^{\alpha-1} (n/s)^{n+1}(d^n/d\lambda^n)(\lambda^\alpha\widehat f(\lambda))|_{\lambda= n/s}\,ds,$ where $$\widehat f$$ is the Laplace transform of the vector-valued function $$f$$. The result is applied to $$\alpha$$-times integrated semigroups resp. cosine functions.
##### MSC:
 44A10 Laplace transform 47D62 Integrated semigroups 44A20 Integral transforms of special functions
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