Ince equation

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

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

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

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

where $C$ is a constant.

See also

• Whittaker–Hill equation
• Ince–Gaussian beam

References

• 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", Journal of Mathematical Physics 16 (3): 512–517, doi:10.1063/1.522574, ISSN 0022-2488, Bibcode: 1975JMP....16..512B, http://researchcommons.waikato.ac.nz/bitstream/10289/1243/1/Kalnins%20variables%207.pdf 
• 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, https://books.google.com/books?id=ML5wm-T4RVQC 
• Mennicken, Reinhard (1968), "On Ince's equation", Archive for Rational Mechanics and Analysis (Springer Berlin / Heidelberg) 29 (2): 144–160, doi:10.1007/BF00281363, ISSN 0003-9527, Bibcode: 1968ArRMA..29..144M 
• Wolf, G. (2010), "Equations of Whittaker–Hill and Ince", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/28.31