| 释义 |
Legendre Math.|ləʒɑ̃dr|
The name of Adrien Marie Legendre (1752–1833), an eminent French mathematician, used attrib. and in the possessive to designate certain expressions investigated by him, esp. (a) the differential equation (1 - x2)d2y/dx2 - 2xdy/dx + n(n + 1)y = 0, (b) its solutions y = Pn(x), where Pn(x) is a polynomial that is the coefficient of hn in the expansion of (1 - 2xh + h2)-1/2, and (c) the associated functions Pnm(x), equal to (1 - x2)m/2 dm / dxm Pn(x).
1875I. Todhunter Elem. Treat. Laplace's Functions i. 1 The coefficient of αn will be a function of x which we shall denote by Pn(x), and shall call Legendre's Coefficient of the nth order. 1880Encycl. Brit. XIII. 21/2 Legendre's function. 1885A. R. Forsyth Treat. Differential Equations v. 152 We have now obtained the complete integral of Legendre's equation in all cases when n is a real constant, by deducing two integrals which are linearly independent..of one another. 1902Encycl. Brit. XXXII. 798/1 (heading) Legendre associated functions. 1930Engineering 26 Dec. 812/3 There are also tables of Legendre functions. 1938S. Dushman Elem. Quantum Mech. vi. 153 The Legendre polynomials form an orthogonal system, since ∫-11Pk(x)Pn(x)dx = 0 (for k ≠ n) [or] = 2/(2k + 1) (for k = n). Ibid. 158 Since the value of any function Pk(cos θ) exhibits k - 1 loops, there are 2(k - 1) circles parallel to the nodal circles at which the function has the same absolute value. It is for this reason that the Legendre coefficients of zero order are known as zonal harmonics. 1953A. D. & K. H. V. Booth Automatic Digital Calculators xvi. 189 The Legendre polynomials..are such that, if any function is expanded in a series of the polynomials, the first n terms of this series will give the best nth degree polynomial approximation to the given function, in the least squares sense. 1962Corson & Lorrain Introd. Electromagn. Fields iv. 172 The solutions of Legendre's equation are called Legendre polynomials, which we denote by Pn(cos θ), there being a different polynomial for each value of the index n. 1970G. K. Woodgate Elem. Atomic Struct. ii. 17 The solutions are proportional to the associated Legendre functions. |