Legendren.
/ləˈʒɒndrə/, U.S. /ləˈʒɑndrə/ Mathematics.
1. Legendre's equation n. (also Legendre equation) any of various equations associated with Legendre; (in later use spec.) = Legendre's differential equation n. at sense 6.
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1817 J. Leslie Elem. Geom. (ed. 3) 295 If you grant the possibility of the triangles, then Legendre's equation will be established.
1885 A. 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.
1934 Math. Gaz. 18 285 On the other hand, it practically ignores the second solution of the Legendre equation.
1962 D. R. Corson & P. Lorrain Introd. Electromagn. Fields iv. 172 The solutions of Legendre's equation are called Legendre polynomials.
2002 R. G. Newton Quantum Physics 343 The coefficient of yl must satisfy Legendre's equation.
2. Legendre function n. (also Legendre's function) any solution of Legendre's differential equation; (in early use spec.) = Legendre polynomial n. at sense 5. In quot. 1840: gamma function n. at gamma n. and adj. Compounds 1b.
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1840 Trans. Royal Soc. Edinb. 14 574 We must, however, make a slight change in the notation, in order to avoid the charge of misapplying Legendre's functions.
1859 London, Edinb. & Dublin Philos. Mag. & Jrnl. Sci. 18 194 P0, P1, Pq, &c. are Legendre's functions.
1873 I. Todhunter Hist. Math. Theories Attraction II. xix. 23 These functions might with propriety be called Legendre's functions when they involve only one variable.
1930 Engineering 26 Dec. 812/3 There are also tables of Legendre functions.
2007 K.-T. Tang Math. Methods for Scientists & Engineers III. iv. 163 The most frequently encountered functions in solving second-order differential equations are trigonometric, hyperbolic, Bessel, and Legendre functions.
3. Legendre symbol n. (also Legendre's symbol) a function, denoted ( a | p ), whose value for a whole number a and an odd prime number p is the remainder left when a to the power ( p − 1 )/2 is divided by p; the value taken by this function for given values of a and p.Whatever the choice of a and p, the value of ( a | p ), written modulo p, is either 1, − 1, or 0. The values 1 and − 1 indicate whether or not a is a quadratic residue of p, while a value of 0 occurs when a is divisible by p.
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1860 Rep. Brit. Assoc. Advancem. Sci. 1859 250 A much simpler rule may be obtained, by the use of his extension of Legendre's symbol to the case when p is not a prime.
1894 Bull. N.Y. Math. Soc. 3 219 There follow Jacobi's generalization of Legendre's symbol and the generalized law of reciprocity.
1940 Science 22 Nov. 481/1 In this way we obtain new identities, which are noteworthy because of the occurrence of the Legendre symbol.
1964 A. H. Beiler Recreations in Theory of Numbers (1966) xix. 201 The Legendre symbol ( r | p )..compactly represents the ‘quadratic character’ of r with respect to the primep.
2014 R. E. Blahut Cryptogr. & Secure Communication ii. 44 The Legendre symbol is not defined for p = 2.
4. Legendre coefficient n. (also Legendre's coefficient) any of various coefficients, appearing in certain mathematical expressions, associated with Legendre; (in early use spec.) = Legendre polynomial n. at sense 5.
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1873 I. Todhunter Hist. Math. Theories Attraction II. xxxiv. 329 Nor does he make any use of the results he obtains with respect to Legendre's coefficients.
1919 Proc. Royal Soc. 1918–19 A. 95 237 This is the case, for example, with series of Legendre coefficients.
1960 Proc. Amer. Math. Soc. 11 818 What is new in this note is the proof that the Legendre coefficients of these expansions have the form indicated above.
2002 P. Attard Thermodynamics & Statist. Mech. ix. 218 It is convenient to evaluate the Legendre coefficients by a discrete, orthogonal technique.
5. Legendre polynomial n. (also Legendre's polynomial) a polynomial solution to the legendre differential equation, first stated by Legendre.For a given value of n, the Legendre polynomial Pn(x) arises as the coefficient of hn in the expansion of (1 − 2xh + h2)−1/2.
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1893 Bull. N.Y. Math. Soc. 3 44 The relation..is shown to define completely the polynomial Pn to a constant factor près—this polynomial is, of course, Legendre's polynomial.
1938 S. Dushman Elem. Quantum Mech. vi. 153 The Legendre polynomials form an orthogonal system.
1974 Proc. Royal Soc. A. 337 50 To solve practical problems, it is also possible..to use well-known functions, e.g. Legendre polynomials or trigonometric functions, as a comparison set.
2013 H. Kaper & H. Engler Math. & Climate xv. 184 The Legendre polynomials provided a convenient coordinate system.
6. Legendre's differential equation n. (also Legendre differential equation) a second order differential equation which frequently arises in physics, such as when modelling the flow of an ideal fluid past a spherical body, a solution to which being found by Legendre in 1782.This equation is often given in the form (1 − x2) d2y/dx2 − 2x dy/dx + n(n + 1)y = 0.
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1902 A. R. Forsyth Theory Differential Equations 160 One such example has already been indicated, in Legendre's differential equation.
1943 Amer. Math. Monthly 50 291 The Legendre functions of the second kind are met with in the problem of solving completely the Legendre differential equation.
1972 Math. Gaz. 56 355 They use Legendre's differential equation to prove the orthogonality of Legendre polynomials.
2010 O. Christensen Functions, Spaces, & Expansions xi. 215 We consider Legendre's differential equation and derive a class of polynomial solutions, the Legendre polynomials.
This entry has been updated (OED Third Edition, March 2016; most recently modified version published online March 2022).
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