  Laplace's method is one of the fundamental techniques in the asymptotic
approximation of integrals. The coefficients appearing in the resulting
asymptotic expansion, arise as the coefficients of a convergent or asymptotic
series of a function defined in an implicit form. Due to the tedious
computation of these coefficients, most standard textbooks on asymptotic
approximations of integrals do not give explicit formulas for them.
Nevertheless, we can find some more or less explicit representations for the
coefficients in the literature: Perron's formula gives them in terms of
derivatives of an explicit function; Campbell, Fr\"oman and Walles simplified
Perron's method by computing these derivatives using an explicit recurrence
relation. The most recent contribution is due to Wojdylo, who rediscovered the
Campbell, Fr\"oman and Walles formula and rewrote it in terms of partial
ordinary Bell polynomials. In this paper, we provide an alternative
representation for the coefficients, which contains ordinary potential
polynomials. The proof is based on Perron's formula and a theorem of Comtet.
The asymptotic expansions of the gamma function and the incomplete gamma
function are given as illustrations.
