A function  which is one of the solutions to the modified Bessel differential equation and is closely related to the Bessel function of the first kind . The above plot shows  for , 2, ..., 5. The modified Bessel function of the first kind is implemented in the Wolfram Language as BesselI[nu, z].

The modified Bessel function of the first kind  can be defined by the contour integral

(1)

where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).

In terms of ,

(2)

For a real number , the function can be computed using

(3)

where  is the gamma function. An integral formula is

(4)

which simplifies for  an integer  to

(5)

(Abramowitz and Stegun 1972, p. 376).

A derivative identity for expressing higher order modified Bessel functions in terms of  is

(6)

where  is a Chebyshev polynomial of the first kind.

The special case of  gives  as the series

(7)

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Modified Bessel Functions  and ." §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,  and ." §11.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 610-616, 1985.

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.

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  and " and "The General Hyperbolic Bessel Function ." Chs. 49-50 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 479-487 and 489-497, 1987.