Ince equation

In mathematics, the Ince equation, named for Edward Lindsay Ince, is the differential equation

${\displaystyle w^{\prime \prime }+\xi \sin(2z)w^{\prime }+(\eta -p\xi \cos(2z))w=0.\,}$

When p is a non-negative integer, it has polynomial solutions called Ince polynomials. In particular, when ${\displaystyle p=1,\eta \pm \xi =1}$, then it has a closed-form solution¹

${\displaystyle w(z)=Ce^{-iz}(e^{2iz}\mp 1)}$

where ${\displaystyle C}$ is a constant.