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Ince Polynomial
In mathematics, the Ince equation, named for Edward Lindsay Ince, 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 *Whittaker–Hill equation *Ince–Gaussian beam References * * * *{{dlmf, id=28.31, title=Equations of Whittaker–Hill and Ince, first=G. , last=Wolf Ordinary differential equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Ince equation, a generalization of the Mathieu equation. Life He was born in Amblecote in Worcestershire on 30 November 1891, the only son of Caroline Clara Cutler and her husband Edward Ince, an Inland Revenue officer. The family moved to Criccieth near Portmadoc in Wales soon after he was born. His family moved to Perth, Scotland, Perth in Scotland around 1901, living at 6 Queens Avenue in the Craigie district, to the south-west of the city. He attended Perth Academy. He studied mathematics at the University of Edinburgh from 1909, graduating in 1913. Failing the medical for World War I service, he won a scholarship and went to the University of Cambridge where he graduated with an MA, and won the Smith's Prize in 1918. That same ye ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with minimal period \pi and average zero. By these we mean that for all t :f(t+\pi)=f(t), and :\int_0^\pi f(t) \,dt=0, and if p is a number with 0 < p < \pi , the equation must fail for some . It is named after , who introduced it in 1886. Because has period , the Hill equation can be rewritten using the of : : |
Ince–Gaussian Beam
In optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse Gaussian mode describes the intended output of many lasers, as such a beam diverges less and can be focused better than any other. When a Gaussian beam is refocused by an ideal lens, a new Gaussian beam is produced. The electric and magnetic field amplitude profiles along a circular Gaussian beam of a given wavelength and polarization are determined by two parameters: the waist , which is a measure of the width of the beam at its narrowest point, and the position relative to the waist.Svelto, pp. 153–5. Since the Gaussian function is infinite in extent, perfect Gaussian beams do not exist in nature, and the edges of any such beam would be cut off by any finite lens or mirror. However, the Gaussian is a useful approxima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Mathematical Physics
The ''Journal of Mathematical Physics'' is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics. The journal was first published bimonthly beginning in January 1960; it became a monthly publication in 1963. The current editor is Jan Philip Solovej from University of Copenhagen The University of Copenhagen (, KU) is a public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in Scandinavia, after Uppsala University. .... Its 2018 Impact Factor is 1.355 Abstracting and indexing This journal is indexed by the following services: 2013. References External ...
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