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 solution[1]

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

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

See also

• Whittaker–Hill equation
• Ince–Gaussian beam

References

1. Cheung, Tsz Yung. "Liouvillian solutions of Whittaker-Ince equation". Journal of Symbolic Computation. 115 (March-April 2023): 18–38. doi:10.1016/j.jsc.2022.07.002.

• Boyer, C. P.; Kalnins, E. G.; Miller, W. Jr. (1975), "Lie theory and separation of variables. VII. The harmonic oscillator in elliptic coordinates and Ince polynomials" (PDF), Journal of Mathematical Physics, 16 (3): 512–517, Bibcode:1975JMP....16..512B, doi:10.1063/1.522574, hdl:10289/1243, ISSN 0022-2488, MR 0372384
• Magnus, Wilhelm; Winkler, Stanley (1966), Hill's equation, Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons\, New York-London-Sydney, ISBN 978-0-486-49565-1, MR 0197830
• Mennicken, Reinhard (1968), "On Ince's equation", Archive for Rational Mechanics and Analysis, 29 (2), Springer Berlin / Heidelberg: 144–160, Bibcode:1968ArRMA..29..144M, doi:10.1007/BF00281363, ISSN 0003-9527, MR 0223636, S2CID 122886716
• Wolf, G. (2010), "Equations of Whittaker–Hill and Ince", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.