Modified Bessel Function of the First Kind

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 $J_n(x)$,

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

For a Real Number $\nu$, the function can be computed using

$$I_\nu(z)=({\textstyle{1\over 2}}z)^\nu\sum_{k=0}^\infty {({\textstyle{1\over 4}}z^2)^k\over k!\Gamma(\nu+k+1)},$$ (2)

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

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

which simplifies for $\nu$ an Integer $n$ to

$$I_n(z) = {1\over\pi} \int_0^\pi e^{z\cos\theta}\cos(n\theta)\,d\theta$$ (4)

(Abramowitz and Stegun 1972, p. 376).

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

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

References

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.'' http://www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html

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.