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  2. Jacobi identity - Wikipedia

    en.wikipedia.org/wiki/Jacobi_identity

    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:

  3. Theta function - Wikipedia

    en.wikipedia.org/wiki/Theta_function

    There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables z and τ, where z can be any complex number and τ is the half-period ratio, confined to the upper half-plane, which means it has a positive ...

  4. Carl Gustav Jacob Jacobi - Wikipedia

    en.wikipedia.org/wiki/Carl_Gustav_Jacob_Jacobi

    Carl Gustav Jacob Jacobi (/ dʒ ə ˈ k oʊ b i /; [2] German:; 10 December 1804 – 18 February 1851) [a] was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory.

  5. Jacobi–Anger expansion - Wikipedia

    en.wikipedia.org/wiki/Jacobi–Anger_expansion

    In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and cylindrical waves), and in signal processing (to describe FM signals).

  6. Jacobi - Wikipedia

    en.wikipedia.org/wiki/Jacobi

    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

  7. Lie coalgebra - Wikipedia

    en.wikipedia.org/wiki/Lie_coalgebra

    Thus, by the double-duality isomorphism (more precisely, by the double-duality monomorphism, since the vector space needs not be finite-dimensional), the Jacobi identity is satisfied. In particular, note that this proof demonstrates that the cocycle condition d 2 = 0 {\displaystyle d^{2}=0} is in a sense dual to the Jacobi identity.

  8. Jacobi triple product - Wikipedia

    en.wikipedia.org/wiki/Jacobi_triple_product

    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.

  9. Courant bracket - Wikipedia

    en.wikipedia.org/wiki/Courant_bracket

    The Courant bracket does not satisfy the Jacobi identity and so it does not define a Lie algebroid, in addition it fails to satisfy the Lie algebroid condition on the anchor map. Instead it defines a more general structure introduced by Zhang-Ju Liu , Alan Weinstein and Ping Xu known as a Courant algebroid .