lommel Differential Equation

The Lommel differential equation is a generalization of the Bessel differential equation given by

(1)

or, in the most general form, by

(2)

The case  is the most common (Watson 1966, p. 345; Zwillinger 1997, p. 125; Gradshteyn and Ryzhik 2000, p. 937), and its solutions are given by

			(3)
			(4)

where  are Lommel functions. Note that  is most commonly written simply as .

The second-order ordinary differential equation

(5)

is sometimes also called the Lommel differential equation

REFERENCES:

Chandrasekhar, S. Radiative Transfer. New York: Dover, p. 369, 1960.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 125, 1997