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!

 

[tacman]

Yes, Borel resummation can be applied to integrals. Borel resummation is a technique used to obtain the sum of a divergent series by applying the Borel transform. The Borel transform converts the series into an integral, which can then be evaluated using the Borel resummation method.

In the case of integrals, the Borel transform is applied to the integrand function, which then allows for the evaluation of the integral using the Borel resummation method. The Borel transform of an integral can be written as the Laplace transform of the integrand function, as shown in the given equation.

The Borel resummation method can be used to evaluate the integral by considering the analytic continuation of the Laplace transform. This allows for the evaluation of the integral at a value of 'u' that is not necessarily an integer, thus extending the applicability of Borel resummation to a wider range of integrals.

Therefore, the Borel resummation method can be applied to integrals, and in some cases, it may provide a more accurate result compared to other summation methods. However, as with any mathematical technique, it is important to carefully consider the assumptions and limitations of the method before applying it to a specific integral.