Lommel Differential Equation
The Lommel differential equation is a generalization of the Bessel differential equation given by
| z^2y^('')+zy^'+(z^2-nu^2)y=kz^(mu+1), |
(1)
|
or, in the most general form, by
| z^2y^('')+zy^'+(z^2-nu^2)y=+/-kz^(mu+1). |
(2)
|
The case k=+1 is the most common (Watson 1966, p. 345; Zwillinger 1997, p. 125; Gradshteyn and Ryzhik 2000, p. 937), and its solutions are given by
y(z) = C_1J_nu(z)+C_2Y_nu(z)+s_(mu,nu)^((1))(z)
(3)
y(z) = C_1J_nu(z)+C_2Y_nu(z)+s_(mu,nu)^((2))(z)
(4)
where s_(mu,nu)^((m))(z) are Lommel functions of the first and second kind for m=1, 2, respectively. Note that s_(mu,nu)^((1))(z) is most commonly written simply as s_(mu,nu)(z).
The second-order ordinary differential equation
| y^('')+g(y)y^('2)+f(x)y^'=0. |
(5)
|
is sometimes also called the Lommel differential equation.
See also
Lommel Function, Lommel Polynomial, Modified Lommel FunctionExplore with Wolfram|Alpha
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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 EquationCite this as:
Weisstein, Eric W. "Lommel Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LommelDifferentialEquation.html