Analytic Continuation of Laplace Transforms by Means of Asymptotic Series

Abstract

Conditions under which a Laplace transform $\mathrm{L\{F(t)\}=f(s)}$ may be analytically continued, by means of an asymptotic expansion of F, outside the half plane of convergence of the Laplace transform integral are investigated. For t > k define $R_N\hspace{0.17em}\mathrm{by}{\mathrm{\hspace{0.17em}F(t)=t}}^{\mathrm{-\beta }}{\mathrm{\{\sum }}_{\mathrm{n=o}}^N{a}_n{t}^{\mathrm{-n}}{\mathrm{+R}}_N\mathrm{(t)\}}$ for some fixed β with Re β < 1 and suppose that F is integrable on [0, k) for some k ≥ 0. First, it is shown that if R_N(t) = O{N!(σ/t)^N+1} uniformly in N and t > k for some σ > 0, then the singular part of f at s = 0 can be determined in terms of a_i. If β is an integer, then in some neighborhood of s = 0 it is shown that $${\mathrm{L\{F(t)\}=s}}^{\text{β−1}}{\displaystyle \sum _{\text{i=0}}^{\mathrm{-\beta }}}{a}_i{\text{Γ(1−i−β)s}}^i\mathrm{+(}\log \mathrm{\hspace{0.17em}s)g(s)+h(s)}$$ , where g and h are analytic at s = 0 and ${\mathrm{g(s)=\sum }}_{\text{i=1}}^{\infty }{\text{(−1)}}^i{a}_{\mathrm{i-\beta }}{s}^{\text{i−1}}\text{/(i−1)}$ !. If β is not an integer, in some neighborhood of s = 0 it is shown that ${\mathrm{L\{F(t)\}=s}}^{\text{β−1}}\mathrm{g(s)+h(s)}$ , where ${\mathrm{g(s)=\sum }}_{\text{i=0}}^{\infty }{a}_i{\text{Γ(1−i−β)s}}^i$ , with g and h analytic at s = 0. Second, if the estimate on R_N(t) holds uniformly in N and in the complex t plane in the region $\mathrm{(|t|>k)\cap }\mathbf{(}|\arg \mathrm{t|<(}\frac{1}{2}\mathrm{\pi )+\lambda }\mathbf{)}$ for some λ > 0, then the analytic continuation of f can be determined in terms of the a_i. For any k′ > k and for |arg s| < π we have ${\mathrm{L\{F(t)\}=\int }}_0^{\mathrm{k\prime }}{e}^{\mathrm{-st}}{\mathrm{F(t)dt+\int }}_0^{\infty }\mathrm{a(t)\Gamma }\mathbf{(}\text{2−β,k′(s+t)}\mathbf{)}{\mathrm{(s+t)}}^{\text{β−2}}\mathrm{\hspace{0.17em}dt}$ , where Γ is the incomplete gamma function and a(t) is the analytic continuation of ${\sum }_{\text{i=0}}^{\infty }{a}_i{t}^i\mathrm{/i}$ !. If k = 0 in the hypotheses, then with a slight further restriction on F(t) one has ${\text{L{F(t)}=Γ(2−β)∫}}_0^{\infty }{\mathrm{a(t)(s+t)}}^{\text{β−2}}\mathrm{\hspace{0.17em}dt}$ . A generalization and application to a problem in nonrelativistic dispersion theory which includes a Coulomb potential are discussed.