On Laplace Transforms Near the Origin

Asymptotic Estimates of the Distribution of Brownian Hitting Time of a Disc

“…In Appendix we shall indicate how to derive this expansion; therein one will also find three Fourier representations of W . In [6] the present author uses the function N(λ) := ∞ λ W (t)dt to express the leading term of the asymptotic formula of the mean number of visited sites of random walk on Z 2 ; N(λ) is called the Ramanujan function in [8] and [1], where asymptotic expansions of a class of functions including it are obtained.…”

“…Acknowledgments. The author wishes to thank the anonymous referee for his carefully reading the original manuscript, providing several valuable comments, and in particular pointing out the paper [8] and, through it, [1].…”

The expected area of the Wiener sausage swept by a disc

“…In this paper we compute the expectation of the area of S (r) t , which we denote by Area(S (r) t ), for Brownian motion conditioned to be at a prescribed point x ∈ R 2 at time t as well as for free (unconditioned) Brownian motion. Define N(λ), called Ramanujan's function ( [2], [29]) or integral ( [7], page 219), by N(λ) = ∞ 0 e −λu (lg u) 2 + π 2 · du u (λ ≥ 0).…”