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Borel's Expansion

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

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

converges. Then the expansion


\begin{displaymath}
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],
\end{displaymath}

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




© 1996-9 Eric W. Weisstein
1999-05-26