Modified Bessel Differential Equation

The second-order ordinary differential equation

(1)

The solutions are the modified Bessel functions of the first and second kinds, and can be written

			(2)
			(3)

where  is a Bessel function of the first kind,  is a Bessel function of the second kind,  is a modified Bessel function of the first kind, and  is modified Bessel function of the second kind.

If , the modified Bessel differential equation becomes

(4)

which can also be written

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). §9.6.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997.