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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On Laplace transforms near the origin


Author: R. Wong
Journal: Math. Comp. 29 (1975), 573-576
MSC: Primary 44A10
MathSciNet review: 0367564
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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 $ \vert\arg s\vert \leqslant \pi /2 - \Delta $, the $ {c_n}$ being constants.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1975-0367564-1
PII: S 0025-5718(1975)0367564-1
Keywords: Laplace transform, asymptotic expansion, Ramanujan function
Article copyright: © Copyright 1975 American Mathematical Society