IJPAM: Volume 71, No. 3 (2011)

ASYMPTOTIC EXPANSION OF MELLIN
TRANSFORMS IN THE COMPLEX PLANE

Avram Sidi
Computer Science Department
Technion - Israel Institute of Technology
Haifa, 32000, ISRAEL


Abstract. In an earlier paper by the author [A. Sidi, SIAM J. Math. Anal., 16 (1985), pp. 896-906], asymptotic expansions for Mellin transforms $\widehat{f}(z)=\int^\infty_0t^{z-1}f(t)\,dt$ as $z\to\infty$, with $z$ real and positive, were derived. In particular, it was shown there that, for certain classes of functions $u_k(t)$, $k=0,1,\ldots,$ that form asymptotic scales as $t\to\infty$, if $f(t)\sim\sum^\infty_{k=0}A_ku_k(t)$ as $t\to\infty$, then $\widehat{f}(z)\sim\sum^\infty_{k=0}A_k\widehat{u}_k(z)$ as $z\to\infty$. In this note, we show that, for two of the cases considered there, $\widehat{f}(z)\sim\sum^\infty_{k=0}A_k\widehat{u}_k(z)$ as $z\to\infty$, also when $z$ is complex and $ \vert\Im z\vert\leq \eta (\Re z)^c$, for some $c\in(0,1)$ and some fixed, but otherwise arbitrary, $\eta>0$.

Received: June 5, 2011

AMS Subject Classification: 30E10, 30E15, 33C10, 41A60, 44A15

Key Words and Phrases: Mellin transforms, asymptotic expansions, modified Bessel functions

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 71
Issue: 3