Jacobin.

/jaˈkəʊbi/

Etymology: < the name of Karl Gustav Jacob Jacobi (1804–51), German mathematician.

Mathematics.

Used attributively and in the possessive to designate concepts introduced by him or arising out of his work, as Jacobi equation n. (also Jacobi's equation) . Jacobi's identity n. (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 polynomial n. (also Jacobi's polynomial) (formerly Jacobi's function) any of a set of polynomials normally written J_n(p, q; x) and equivalent to F(-n, p+n; q; x), where n is a positive integer and F is the hypergeometric function.

ΘΚΠ

the world > relative properties > number > algebra > [noun] > expression > consisting of specific number of terms

binomial1557

binomy1571

trinomy1571

quadrinomial1673

multinomiala1690

polynomiala1690

trinomiala1690

monomial1706

nomial1717

monome1736

infinitinomial1763

polynome1828

mononomial1844

quantic1854

form1859

Jacobi polynomial1882

Jacobi's function1882

ternariant1882

triquaternion1902

term1957

arity1968

the world > relative properties > number > algebra > [noun] > expression > equation

equation1570

cardanic equation1684

binomial equation1814

simultaneous equation1816

characteristic equation1828

characteristic equation1841

characteristic equation1849

intrinsic equation of a curve1849

complete primitive1859

primitive1862

Poisson's equation1873

Jacobi equation1882

formulaic equation1884

adjoint1889

recursion formula1895

characteristic equation1899

characteristic equation1900

Pell equation1910

Lotka–Volterra equations1937

Langevin equation1943

1882 Q. Jrnl. Pure & Appl. Math. 18 66 (heading) Reduction of the elliptic integrals ∫^dz/ _{(z³ − 1)√(z³ − b³)} and ∫^zdz/ _{(z³ − 1)√(z³ − b³)} to Jacobi's functions.

1886 G. S. Carr Synopsis Elem. Results Math. II. Index 913/1 Polynomials of two variables analogous to Jacobi's.

1889 Cent. Dict. at Equation Jacobi's equation, the equation (ax + by + cz) (ydz − zdy) + (a′x + b′y + c′z) (zdx − xdz) + (a″x + b″y + c″z) (xdy − ydx) = 0.

1902 Encycl. Brit. XXIX. 125/1 If X_i, X_j, X_k are any three linear operators, the identity (known as Jacobi's) (X_i (X_jX_k)) + (X_j(X_kX_i)) + (X_k(X_iX_j)) = 0 holds among them.

1925 Japanese Jrnl. Math. 2 1 The polynomial solution Pn(x), with the leading coefficient 1, of the differential equation (1−x²)y″ + 2[α−β−(α+β)x]y′ + n[n−1+2(α+β)]y = 0 is the so-called Jacobi's polynomial.

1927 E. 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.

1933 L. 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.

1965 E. M. Patterson & D. E. 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.

1971 Amer. Jrnl. Physics 39 501/2 The {T_j (ξ)} are identified with a set of classical orthogonal polynomials, the Jacobi polynomials {P_j ^{(-1/2, 1/2)} (ξ)}.

1986 P. C. West Introd. Supersymmetry & Supergravity ii. 7 As in a Lie algebra we have some generalized Jacobi identities.

This entry has not yet been fully updated (first published 1993; most recently modified version published online September 2020).

单词	jacobi
释义	Jacobin. /jaˈkəʊbi/ Etymology: < the name of Karl Gustav Jacob Jacobi (1804–51), German mathematician. Mathematics. Used attributively and in the possessive to designate concepts introduced by him or arising out of his work, as Jacobi equation n. (also Jacobi's equation) . Jacobi's identity n. (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 polynomial n. (also Jacobi's polynomial) (formerly Jacobi's function) any of a set of polynomials normally written J_n(p, q; x) and equivalent to F(-n, p+n; q; x), where n is a positive integer and F is the hypergeometric function. ΘΚΠ the world > relative properties > number > algebra > [noun] > expression > consisting of specific number of terms binomial1557 binomy1571 trinomy1571 quadrinomial1673 multinomiala1690 polynomiala1690 trinomiala1690 monomial1706 nomial1717 monome1736 infinitinomial1763 polynome1828 mononomial1844 quantic1854 form1859 Jacobi polynomial1882 Jacobi's function1882 ternariant1882 triquaternion1902 term1957 arity1968 the world > relative properties > number > algebra > [noun] > expression > equation equation1570 cardanic equation1684 binomial equation1814 simultaneous equation1816 characteristic equation1828 characteristic equation1841 characteristic equation1849 intrinsic equation of a curve1849 complete primitive1859 primitive1862 Poisson's equation1873 Jacobi equation1882 formulaic equation1884 adjoint1889 recursion formula1895 characteristic equation1899 characteristic equation1900 Pell equation1910 Lotka–Volterra equations1937 Langevin equation1943 1882 Q. Jrnl. Pure & Appl. Math. 18 66 (heading) Reduction of the elliptic integrals ∫^dz/ _{(z³ − 1)√(z³ − b³)} and ∫^zdz/ _{(z³ − 1)√(z³ − b³)} to Jacobi's functions. 1886 G. S. Carr Synopsis Elem. Results Math. II. Index 913/1 Polynomials of two variables analogous to Jacobi's. 1889 Cent. Dict. at Equation Jacobi's equation, the equation (ax + by + cz) (ydz − zdy) + (a′x + b′y + c′z) (zdx − xdz) + (a″x + b″y + c″z) (xdy − ydx) = 0. 1902 Encycl. Brit. XXIX. 125/1 If X_i, X_j, X_k are any three linear operators, the identity (known as Jacobi's) (X_i (X_jX_k)) + (X_j(X_kX_i)) + (X_k(X_iX_j)) = 0 holds among them. 1925 Japanese Jrnl. Math. 2 1 The polynomial solution Pn(x), with the leading coefficient 1, of the differential equation (1−x²)y″ + 2[α−β−(α+β)x]y′ + n[n−1+2(α+β)]y = 0 is the so-called Jacobi's polynomial. 1927 E. 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. 1933 L. 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. 1965 E. M. Patterson & D. E. 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. 1971 Amer. Jrnl. Physics 39 501/2 The {T_j (ξ)} are identified with a set of classical orthogonal polynomials, the Jacobi polynomials {P_j ^{(-1/2, 1/2)} (ξ)}. 1986 P. C. West Introd. Supersymmetry & Supergravity ii. 7 As in a Lie algebra we have some generalized Jacobi identities. This entry has not yet been fully updated (first published 1993; most recently modified version published online September 2020). < n.1882
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