Ince equation explained

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

w ^{\prime\prime} + \xi\sin( 2 z)w ^{\prime} + ( η- p\xi\cos( 2 z))w =0 .

When p is a non-negative integer, it has polynomial solutions called Ince polynomials. In particular, when

p =1 , η \pm\xi =1 , then it has a closed-form solution^[1] w(z) = Ce ^{-iz} (e ^{2iz} \mp 1 )

where

C is a constant.

See also

• Whittaker–Hill equation
• Ince–Gaussian beam

Notes and References

1. Cheung . Tsz Yung . Liouvillian solutions of Whittaker-Ince equation . Journal of Symbolic Computation . 115 . March-April 2023 . 18-38 . 10.1016/j.jsc.2022.07.002.