On Laplace transforms near the origin
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- by R. Wong PDF
- Math. Comp. 29 (1975), 573-576 Request permission
Abstract:
Let $f(t)$ be locally integrable on $[0,\infty )$ and let $L\{ f\} (s)$ denote the Laplace transform of $f(t)$. In this note, we prove that if $f(t) \sim {t^{ - \beta }}\Sigma _{n = 0}^\infty {a_n}{(\log t)^{ - n}}$ as $t \to \infty$, where $0 \leqslant \operatorname {Re} \beta < 1$, then $L\{ f\} (s) \sim {s^{\beta - 1}}\Sigma _{n = 0}^\infty {c_n}{(\log 1/s)^{ - n}}$ as $s \to 0$ in $|\arg s| \leqslant \pi /2 - \Delta$, the ${c_n}$ being constants.References
- C. J. Bouwkamp, Note on an asymptotic expansion, Indiana Univ. Math. J. 21 (1971/72), 547–549. MR 291703, DOI 10.1512/iumj.1971.21.21043
- Richard A. Handelsman and John S. Lew, Asymptotic expansion of Laplace transforms near the origin, SIAM J. Math. Anal. 1 (1970), 118–130. MR 259504, DOI 10.1137/0501012
- R. Wong, On a Laplace integral involving logarithms, SIAM J. Math. Anal. 1 (1970), 360–364. MR 282129, DOI 10.1137/0501033
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Math. Comp. 29 (1975), 573-576
- MSC: Primary 44A10
- DOI: https://doi.org/10.1090/S0025-5718-1975-0367564-1
- MathSciNet review: 0367564