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Modified Bessel Function of the First Kind

\begin{figure}\begin{center}\BoxedEPSF{BesselI.epsf scaled 1000}\end{center}\end{figure}

A function $I_n(x)$ which is one of the solutions to the Modified Bessel Differential Equation and is closely related to the Bessel Function of the First Kind $J_n(x)$. The above plot shows $I_n(x)$ for $n=1$, 2, ..., 5. In terms of $J_n(x)$,

I_n(x) \equiv i^{-n}J_n(ix) = e^{-n\pi i/2}J_n(xe^{i\pi/2}).
\end{displaymath} (1)

For a Real Number $\nu$, the function can be computed using
I_\nu(z)=({\textstyle{1\over 2}}z)^\nu\sum_{k=0}^\infty {({\textstyle{1\over 4}}z^2)^k\over k!\Gamma(\nu+k+1)},
\end{displaymath} (2)

where $\Gamma(z)$ is the Gamma Function. An integral formula is

I_\nu(z) = {1\over \pi} \int_0^\pi{e^{z\cos\theta}\cos(\nu\t...
...-{\sin(\nu\pi)\over\pi} \int_0^\infty e^{-z\cosh t-\nu t}\,dt,
\end{displaymath} (3)

which simplifies for $\nu$ an Integer $n$ to
I_n(z) = {1\over\pi} \int_0^\pi e^{z\cos\theta}\cos(n\theta)\,d\theta
\end{displaymath} (4)

(Abramowitz and Stegun 1972, p. 376).

A derivative identity for expressing higher order modified Bessel functions in terms of $I_0(x)$ is

I_n(x)=T_n\left({d\over dx}\right)I_0(x),
\end{displaymath} (5)

where $T_n(x)$ is a Chebyshev Polynomial of the First Kind.

See also Bessel Function of the First Kind, Modified Bessel Function of the First Kind, Weber's Formula


Abramowitz, M. and Stegun, C. A. (Eds.). ``Modified Bessel Functions $I$ and $K$.'' §9.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 374-377, 1972.

Arfken, G. ``Modified Bessel Functions, $I_\nu(x)$ and $K_\nu(x)$.'' §11.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 610-616, 1985.

Finch, S. ``Favorite Mathematical Constants.''

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions.'' §6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 234-245, 1992.

Spanier, J. and Oldham, K. B. ``The Hyperbolic Bessel Functions $I_0(x)$ and $I_1(x)$'' and ``The General Hyperbolic Bessel Function $I_\nu(x)$.'' Chs. 49-50 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 479-487 and 489-497, 1987.

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© 1996-9 Eric W. Weisstein