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It was introduced by Jacobi in his work Fundamenta Nova Theoriae Functionum Ellipticarum. The Jacobi triple product identity is the Macdonald identity for the affine root system of type A 1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra.
Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a derivation. That form of the Jacobi identity is also used to define the notion of Leibniz algebra. Another rearrangement shows that the Jacobi identity is equivalent to the following identity between the operators of the adjoint representation:
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, ... which is the Jacobi identity for the cross product. Another useful ...
The book introduces Jacobi elliptic functions and the Jacobi triple product identity. One of the most exciting moments of my life was when, after computing several of these series, I went down to our mathematical library and found some of them in Jacobi's "Fundamenta nova theoriae..."[3], with the same coefficients down to the last decimal digit!
In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald ().They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identities found by Dyson (1972), and a 10-fold product identity found by Winquist (1969).
The pentagonal number theorem occurs as a special case of the Jacobi triple product. Q-series generalize Euler's function, which is closely related to the Dedekind eta function, and occurs in the study of modular forms.
In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties.In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta.
Jacobi coordinates, a simplification of coordinates for an n-body system; Jacobi identity for non-associative binary operations; Jacobi's formula for the derivative of the determinant of a matrix; Jacobi triple product, an identity in the theory of theta functions; Jacobi's theorem (disambiguation), several theorems