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  2. Symmetric function - Wikipedia

    en.wikipedia.org/wiki/Symmetric_function

    Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric -tensors on a vector space is isomorphic to the space of homogeneous polynomials of degree on . Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.

  3. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.

  4. Proofs of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_trigonometric...

    For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin θ < θ. So we have ⁡ < <. For negative values of θ we have, by the symmetry of the sine function

  5. Symmetry in mathematics - Wikipedia

    en.wikipedia.org/wiki/Symmetry_in_mathematics

    Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. [1] Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure.

  6. Symmetric polynomial - Wikipedia

    en.wikipedia.org/wiki/Symmetric_polynomial

    One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field.These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots.

  7. Lemniscate elliptic functions - Wikipedia

    en.wikipedia.org/wiki/Lemniscate_elliptic_functions

    Just as for the trigonometric functions, values of the lemniscate functions can be computed for divisions of the lemniscate into n parts of equal length, using only basic arithmetic and square roots, if and only if n is of the form = where k is a non-negative integer and each p i (if any) is a distinct Fermat prime.

  8. Ring of symmetric functions - Wikipedia

    en.wikipedia.org/wiki/Ring_of_symmetric_functions

    The name "symmetric function" for elements of Λ R is a misnomer: in neither construction are the elements functions, and in fact, unlike symmetric polynomials, no function of independent variables can be associated to such elements (for instance e 1 would be the sum of all infinitely many variables, which is not defined unless restrictions are ...

  9. Symmetrically continuous function - Wikipedia

    en.wikipedia.org/wiki/Symmetrically_continuous...

    The set of the symmetrically continuous functions, with the usual scalar multiplication can be easily shown to have the structure of a vector space over , similarly to the usually continuous functions, which form a linear subspace within it.