Lommel function

A Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation:

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

A solution is given by the Lommel function

$s_{\mu ,\nu }(z)={\frac {1}{2}}\pi \left[Y_{\gamma }(z)\int _{0}^{z}z^{\mu }J_{\gamma }(z)\,dz-J_{\gamma }(z)\int _{0}^{z}z^{\mu }Y_{\gamma }(z)\,dz\right]$

where J_γ(z) is a Bessel function of the first kind, and Y_γ(z) a 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

• Anger function
• Lommel polynomial
• Struve function
• Weber function

References

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