From Wikipedia, the free encyclopedia
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.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5581ccae8ac65fd6793a50a0115da5c28329bf84)
When p is a non-negative integer, it has polynomial solutions called Ince polynomials. In particular, when
, then it has a closed-form solution
[1]
where
is a constant.
See also
References
-
^ 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.