Ince Polynomial

	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 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 ... [...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. He then bega ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Differential Equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...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 by minimal period \pi . 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 $f(t+p) = f(t)$ must fail for some $t$ . It is named after George William Hill, who introduced it in 1886. Because $f(t)$ has period $\pi$ , the Hill equation can be rewritten using the Fourier series of $f(t)$ : : $\frac+\left(\theta_0+2\sum_^\infty \theta_n \cos(2nt)+\sum_^\infty \phi_m \sin(2mt) \right ) y=0.$ Important special cases of Hill's equation include the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 fundamental (or TEM00) transverse Gaussian mode describes the intended output of most (but not all) lasers, as such a beam can be focused into the most concentrated spot. When such a beam is refocused by a lens, the transverse ''phase'' dependence is altered; this results in a ''different'' Gaussian beam. The electric and magnetic field amplitude profiles along any such circular Gaussian beam (for a given wavelength and polarization) are determined by a single parameter: the so-called waist . At any position relative to the waist (focus) along a beam having a specified , the field amplitudes and phases are thereby determinedSvelto, pp. 153–5. as detailed below. The equations below assume a beam with a circular cross-section at all va ... [...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 ( da, Københavns Universitet, KU) is a prestigious public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in .... Its 2018 Impact Factor is 1.355 Abstracting and indexing This journal is indexed by the following services:Wellesley College Library 2013. References External links [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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