Can Borel resummation be applied to integrals ?

[Klaus_Hoffmann]

The question is can we obtain the 'Borel sum' of an integral of f(x) from 0 to infinity as the Laplace transform of

\int_{0}^{\infty}dx \frac{f(x)}{\Gamma(x+1+u)t^{x+u}

where 'alpha' is a real or Complex number

 

[Klaus_Hoffmann]

i meant (but the nasty latex does not work)

\int_{0}^{\infty} dx \frac{f(x)}{G(x+1+u)}t^{x+u}

then the Laplace transform evaluated at s=1 is the 'Borel sum' of the integral

\int_{0}^{\infty} dx f(x)

Of course G(x) is the 'Gamma function' generalization of factorial n!