Lommel function

A Lommel differential equation, named after Eugen von Lommel, is an ordinary differential equation that generalizes the Bessel differential equation:

$z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}-(z^{2}-v^{2})y=kz^{\mu +1}.$

A further generalization yields:

$z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}-(z^{2}-v^{2})y=\pm kz^{\mu +1}.$

The solutions for these are given by the Lommel functions

$y={\frac {1}{2}}\pi k\lbrack Y_{\gamma }(z)\int _{0}^{z}Z^{\mu }J_{\gamma }(z)dz-J_{\gamma }(z)\int _{0}^{z}Z^{\mu }Y_{\gamma }(z)dz\rbrack$

for $k>0$, where $J_{\gamma }(z$) is a Bessel function of the first kind, and $Y_{\gamma }(z$) a Bessel function of the second kind. For $k<0$ the Lommel function becomes

$y=I_{\gamma }(z)k\lbrack Y_{\gamma }(z)\int _{0}^{z}Z^{\mu }K_{\gamma }(z)dz-J_{\gamma }(z)\int _{0}^{z}Z^{\mu }I_{\gamma }(z)dz\rbrack$

where $K_{\gamma }(z$) is a modified Bessel function of the first kind, and $I_{\gamma }(z$) a modified Bessel function of the second kind.

The second-order ordinary differential equation

$y''+g(y){y'}^{2}+f(x)y'=0\,$

is sometimes also called the Lommel differential equation.

See also

• Lommel polynomial

References

• Weisstein, Eric W. "Lommel Differential Equation." From MathWorld--A Wolfram Web Resource.