| 释义 |
Jacobi, n. Math.
(ˈdʒækəbɪ, ‖ jaˈkoːbi)
The name of Karl Gustav Jacob Jacobi (1804–51), German mathematician, used attrib. and in the possessive to designate concepts introduced by him or arising out of his work, as Jacobi('s) equation, identity, (a) the identity [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 , where A, B, and C are any linear operators and square brackets denote the taking of the commutator of two operators; (b) any of various other identities that may be expressed in a typographically similar way; Jacobi('s) polynomial (formerly Jacobi's function), any of a set of polynomials normally written Jn(p, q; x) and equivalent to F(-n, p+n; q; x) , where n is a positive integer and F is the hypergeometric function.
1882Q. Jrnl. Pure & Appl. Math. XVIII. 66 (heading) Reduction of the elliptic integrals ∫ dz / (z3 - 1)√(z3 - b3) and ∫ zdz / (z3 - 1)√(z3 - b3) to Jacobi's functions. 1886G. S. Carr Synopsis Elem. Results Pure Math. 913/1 (Index), Polynomials of two variables analogous to Jacobi's. 1889Cent. Dict. s.v. Equation, Jacobi's equation, the equation (ax + by + cz) (ydz - zdy) + (a′x + b′y + c′z) (zdx - xdz) + (a{pp}x + b{pp}y + c{pp}z) (xdy - ydx) = 0 . 1902Encycl. Brit. XXIX. 125/1 If Xi, Xj, Xk are any three linear operators, the identity (known as Jacobi's) (Xi (XjXk)) + (Xj(XkXi)) + (Xk(XiXj)) = 0 holds among them. 1925Jap. Jrnl. Math. II. 1 The polynomial solution Pn(x) , with the leading coefficient 1, of the differential equation (1-x2)y{pp} + 2[α-β-(α+β)x]y′ + n[n-1+2(α+β)]y = 0 is the so-called Jacobi's polynomial. 1927E. L. Ince Ordinary Differential Equations ii. 22 The Jacobi equation, (a1 + b1x + c1y) (xdy - ydx) - (a2 + b2x + c2y)dy + (a3 + b3x + c3y)dx = 0, in which the coefficients a, b, c are constants . 1933L. P. Eisenhart Continuous Groups of Transformations vi. 250 For any three functions u, v, w..the following equation is an identity ((u, v), w) + ((v, w), u) + ((w, u), v) = 0 . It is called the Jacobi identity. 1965Patterson & Rutherford Elem. Abstr. Algebra v. 198 We assume that, for all vectors a, b, c, we have a × (b × c) + b × (c × a) + c × (a × b) = 0 ..which is known as Jacobi's identity. 1971Amer. Jrnl. Physics XXXIX. 501/2 The {ob}Tj (ξ){cb} are identified with a set of classical orthogonal polynomials, the Jacobi polynomials {ob}Pj (-1/2, 1/2) (ξ){cb}. 1986P. C. West Introd. Supersymmetry & Supergravity ii. 7 As in a Lie algebra we have some generalized Jacobi identities. |