Borel's Expansion

Let $\phi(t)=\sum_{n=0}^\infty A_nt^n$ be any function for which the integral

$$I(x)\equiv \int_0^\infty e^{-tx}t^p\phi(t)\,dt$$

converges. Then the expansion

$$I(x)={\Gamma(p+1)\over x^{p+1}} \left[{A_0+(p+1){A_1\over x}+ (p+1)(p+2){A_2\over x^2}+\ldots}\right],$$

where $\Gamma(z)$ is the Gamma Function, is usually an Asymptotic Series for $I(x)$.