An Explicit Formula for the Coefficients in Laplace’s Method

Abstract

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, Fröman 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, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that 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.

Appendix:  Powers of Power Series

Let $F ( x ) = 1 + \sum_{n = 1}^{\infty}{f_{n} x^{n} }$ be a formal power series. For any nonnegative integer k, we define the partial ordinary Bell polynomials B _n,k (f ₁,f ₂,…,f _n−k+1) (associated with F) by the generating function

$$\bigl( {F ( x ) - 1} \bigr)^k = \Biggl( {\sum _{n = 1}^\infty {f_n x^n } } \Biggr)^k = \sum _{n = k}^\infty {\mathsf {B}_{n,k} ( {f_1 ,f_2 , \ldots,f_{n - k + 1} } )x^n }, $$

so that B _0,0=1, B _n,0=0 (n≥1), B _n,1=f _n , and $\mathsf{B}_{n,2} = \sum\nolimits_{j = 1}^{n-1} {f_{j} f_{n - j} }$. From the simple identity, (F(x)−1)^k+1=(F(x)−1)(F(x)−1)^k, we obtain the recurrence relation

$$ \mathsf{B}_{n,k + 1} ( {f_1 ,f_2 , \ldots,f_{n - k} } ) = \sum _{j = 1}^{n - k} {f_j \mathsf{B}_{n - j,k} ( {f_1 ,f_2 , \ldots,f_{n - j - k + 1} } )} . $$

(A.1)

An explicit representation, that follows from the definition, is given by

$$\mathsf{B}_{n,k} ( {f_1 ,f_2 , \ldots,f_{n - k + 1} } ) = \sum{\frac{{k!}}{{k_1 !k_2 ! \cdots k_{n - k + 1} !}}f_1^{k_1 } f_2^{k_2 } \cdots f_{n - k + 1}^{k_{n - k + 1} } }, $$

where the sum runs over all sequences k ₁,k ₂,…,k _n of nonnegative integers such that k ₁+2k ₂+⋯+(n−k+1)k _n−k+1=n and k ₁+k ₂+⋯+k _n−k+1=k.

For any complex number ρ, we define the ordinary potential polynomials A _ρ,n (f ₁,f ₂,…,f _n ) (associated with F) by the generating function

$$\bigl( {F ( x )} \bigr)^\rho = \Biggl( {1 + \sum _{n = 1}^\infty {f_n x^n } } \Biggr)^\rho = \sum _{n = 0}^\infty {\mathsf{A}_{\rho,n} ( {f_1 ,f_2 , \ldots,f_n } )x^n }, $$

hence,

$$\mathsf{A}_{\rho,n} ( {f_1 ,f_2 , \ldots,f_n } ) = \sum _{k = 0}^n { \binom{\rho}{k} \mathsf{B}_{n,k} ( {f_1 ,f_2 , \ldots,f_{n - k + 1} } )}. $$

The first few are A _ρ,0=1, A _ρ,1=ρf ₁, $\mathsf{A}_{\rho,2} = \rho f_{2} + \binom{\rho}{2}f_{1}^{2}$, and, in general,

$$\mathsf{A}_{\rho,n} ( {f_1 ,f_2 , \ldots,f_n } ) = \sum {\binom{\rho}{k} \frac{{k!}}{{k_1 !k_2 ! \cdots k_n !}}f_1^{k_1 } f_2^{k_2 } \cdots f_n^{k_n } } , $$

where the sum extends over all sequences k ₁,k ₂,…,k _n of nonnegative integers such that k ₁+2k ₂+⋯+nk _n =n and k ₁+k ₂+⋯+k _n =k. Since (F(x))^ρ=(F(x))^ρ−1 F(x), we have the recurrence

$$ \mathsf{A}_{\rho,n} ( {f_1 ,f_2 , \ldots,f_n } ) = \mathsf {A}_{\rho - 1,n} ( {f_1 ,f_2 , \ldots,f_n } ) + \sum _{k = 1}^n {f_k \mathsf{A}_{\rho - 1,n - k} ( {f_1 ,f_2 , \ldots ,f_{n - k} } )} . $$

(A.2)

In general, if $G ( x ) = \sum\nolimits_{n = 0}^{\infty}{g_{n} x^{n} }$ is a formal power series, then by definition, we have

$$G \bigl( {y \bigl( {F ( x ) - 1} \bigr)} \bigr) = \sum _{n = 0}^\infty { \Biggl( {\sum _{k = 0}^n {g_k \mathsf{B}_{n,k} ( {f_1 ,f_2 , \ldots,f_{n - k + 1} } )y^k } } \Biggr)x^n } . $$

Specially,

$$\exp \Biggl( {y\sum _{n = 1}^\infty {f_n x^n } } \Biggr) = \sum _{n = 0}^\infty { \Biggl( {\sum _{k = 0}^n {\frac{{\mathsf {B}_{n,k} ( {f_1 ,f_2 , \ldots,f_{n - k + 1} } )}}{{k!}}y^k } } \Biggr)x^n } . $$

For more details see, e.g., Comtet’s book [3, pp. 133–153].