Ince Polynomial
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In mathematics, the Ince equation, named for
Edward Lindsay Ince Prof Edward Lindsay Ince FRSE (30 November 1891 – 16 March 1941) was a British mathematician who worked on differential equations, especially those with periodic coefficients such as the Mathieu equation and the Lamé equation. He introduced the ...
, is the differential equation :w^+\xi\sin(2z)w^+(\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)=Ce^(e^\mp 1) where C is a constant.


See also

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Whittaker–Hill equation In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation : \frac + f(t) y = 0, where f(t) is a periodic function by minimal period \pi . By these we mean that for all t :f(t+\ ...
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Ince–Gaussian beam In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This ...


References

* * * *{{dlmf, id=28.31, title=Equations of Whittaker–Hill and Ince, first=G. , last=Wolf Ordinary differential equations