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
where 
are Lommel functions of the first and second kind
for 
,
2, respectively. Note that 
is most commonly written simply as 
.
The second-order ordinary
differential equation
 |
(5)
|
is sometimes also called the Lommel differential equation.
See also
Lommel Function,
Lommel
Polynomial,
Modified Lommel Function
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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.Referenced on Wolfram|Alpha
Lommel Differential Equation
Cite this as:
Weisstein, Eric W. "Lommel Differential Equation."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LommelDifferentialEquation.html
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