IJPAM: Volume 71, No. 3 (2011)

ASYMPTOTICS OF MODIFIED BESSEL
FUNCTIONS OF HIGH ORDER

Avram Sidi$^1$, Philip E. Hoggan$^2$
$^1$Computer Science Department
Technion - Israel Institute of Technology
Haifa, 32000, ISRAEL
$^2$LASMEA, UMR 6602 CNRS
Université Blaise Pascal
24, Avenue des Landais, 63177, Aubiere Cedex, FRANCE


Abstract. In this work, we present two sets of full asymptotic expansions for the modified Bessel functions $I_\nu(z)$ and $K_\nu(z)$ and a full asymptotic expansion for $I_\nu(z)K_\nu(z)$ as $\nu\to\infty$ and $z$ is fixed with $\vert\arg z\vert<\pi$. In particular, we show that

\begin{displaymath} I_\nu(z)\sim\frac{(\frac{1}{2}z)^\nu}{\Gamma(\nu+1)} \sum^\i... ...\text{as $\nu\to\infty$,}\quad\vert\arg\nu\vert\leq\pi-\delta ,\end{displaymath}

and

\begin{displaymath}K_\nu(z)\sim\frac{1}{2}\,\frac{\Gamma(\nu)}{(\frac{1}{2}z)^\n... ...u\to\infty$,}\quad \vert\arg\nu\vert\leq\tfrac{1}{2}\pi-\delta,\end{displaymath}

where, for each $m$, $b_m(z)$ is a polynomial of degree $m$ in $z^2$, whose coefficients alternate in sign. Actually,

\begin{displaymath}b_0(z)=1;\quad b_m(z)=\sum^m_{k=1}(-1)^{m-k}\frac{S(m,k)}{k!} (\tfrac{1}{4}z^2)^k,\quad m=0,1,\ldots,\end{displaymath}

where $S(m,k)$ are the Stirling numbers of the second kind. We also compare the asymptotic expansions of this work with those existing in the literature.

Received: June 5, 2011

AMS Subject Classification: 33C10, 34E05, 35C20, 41A60

Key Words and Phrases: modified Bessel functions, high order, asymptotic expansions

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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