\label{original}$$ then its Borel transform is defined by $$\widetilde{f}( t ) \;=\; a_0 \delta(t) \;+\; \sum_{n=1}^{\infty}
 { 1 \over (n-1)! }\;  a_{n} t^{n-1} \;.
\label{borel}$$ In this paper, we indicate the Borel transform of a quantity $f$ either as $\tilde f$ or $B[f]$. Clearly, the series (\[borel\]) for the Borel transform has better convergence properties than the original power series (\[original\]). The function $f(\alpha_s)$ that generates the series (\[original\]) can be recovered from the Borel transform $\widetilde{f}(t)$ through the inverse Borel transform $$f(\alpha_s) \;=\; \int_0^{\infty} \;dt \; e ^{-t/\alpha_s} \; \widetilde{f} 
(t)\;,
\label{borel-inverse}$$ provided that the integral (\[borel-inverse\]) is sufficiently convergent and that the Borel transform (\[borel\]) is well-defined along the positive real axis. [^1]

The Borel transform of the series (\[bubbles\]) is $$\widetilde{\alpha}_s^{(1)}(t)= {(\mu^2/e^C)^u\over l^{2u}},
\label{bubbles-tx}$$ where $u=\beta_0t$. If we insert the Borel transform (\[bubbles-tx\]) into the one-loop integral (\[one-loop\]), we obtain $$(\mu^2/e^C)^u\int_0^\Lambda dl\, l^{m-2u}
=(\mu^2/e^C)^u{\Lambda^{m+1-2u}\over m+1-2u},
\label{bubble-integral}$$ where $\Lambda$ is an ultraviolet cutoff, and, in the spirit of dimensional regularization, we have defined the integral for ${\rm
Re}~u>(m+1)/2$ by analytic continuation. The integral has a pole in the complex $u$ plane at $u=(m+1)/2$. This singularity appears even though the original one-loop integral is perfectly well-behaved. A pole in 